#72235809

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14673
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂⁶	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

2037 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,14,7,33,5,29)(3,16,20,35,34,47,58,39)(4,22,26,43,27,45,25,46)(8,15,30,49,9,11,31,13)(10,17,38,19,56,21,37,32)(12,40,44,24,42,62,48,28)(18,52,55,60,53,50,59,64)(23,61,57,36,51,63,54,41);(1,3,5,19,7,34,6,32)(2,10,13,37,33,56,49,38)(4,23,25,59,27,51,26,55)(8,21,31,58,9,17,30,20)(11,39,14,47,15,35,29,16)(12,41,43,63,42,36,46,61)(18,40,54,28,53,62,57,24)(22,60,44,52,45,64,48,50);(2,11)(3,17)(4,24)(5,30)(6,31)(10,35)(12,42)(15,33)(16,47)(18,53)(21,34)(22,45)(23,51)(25,26)(27,28)(37,38)(39,56)(40,62)(43,48)(44,46)(50,61)(52,63)(54,59)(55,57);(1,4,9,28)(2,12,11,22)(3,18,17,51)(5,26,30,40)(6,25,31,62)(7,27,8,24)(10,36,39,64)(13,46,14,48)(15,45,33,42)(16,50,38,61)(19,57,20,59)(21,23,34,53)(29,44,49,43)(32,54,58,55)(35,60,56,41)(37,63,47,52)

Polycyclic presentation:

2^10|a^2=f,b^2=e,d^2=j,e^2=ij,f^2=ij,b^a=bfg,c^a=cgij,c^b=cgh,d^a=dhj,d^b=dgi,d^c=di,e^a=ei,e^c=ej,e^d=eij,f^b=fi,f^c=fj,f^d=fij,g^a=gj,g^b=gij,h^a=hij,h^b=hi

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235822, #72235838, #72235857
maximal normal subgroup of	#72235822, #72235838, #72235857

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);