#72303717

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#11174
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#123590
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

4989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=j,f^2=j,g^2=j,b^a=begj,c^a=cd,c^b=cdfj,d^a=dj,e^b=ej,e^c=ej,f^a=fi,f^b=fij,f^c=fj,g^b=gij,g^c=gij,g^f=gj,h^a=hij,h^c=hi,h^f=hj,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303733, #72312305, #72312878
maximal normal subgroup of	#72303733, #72312305, #72312878

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.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);