#72320876

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#366548
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#7622837	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ C₂×#1757 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3394 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efij,c^2=eij,f^2=i,i^2=j,b^a=bf,c^a=cfh,c^b=cfj,d^a=dg,d^b=dj,d^c=dj,e^b=ej,e^c=ej,f^a=fi,f^b=fj,f^c=fi,g^c=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320921, #72321054, #72343936
maximal normal subgroup of	#72320921, #72321054, #72343936

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