#72280321

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#25549
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#403956
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2932 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4299 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)(3,4)(5,27,30,56)(6,33)(7,11)(8,39)(9,15)(10,31,45,61)(12,38)(13,36)(14,35)(16,48,42,62)(17,37)(18,24)(19,23)(20,47)(21,26)(22,50,52,63)(25,40)(28,55,49,53)(29,58,60,46)(32,44,59,43)(34,54)(41,51,64,57);(1,3)(2,4)(5,28,30,49)(6,34)(7,18)(8,40)(9,21)(10,41,45,64)(11,24)(12,37)(13,23)(14,47)(15,26)(16,29,42,60)(17,38)(19,36)(20,35)(22,43,52,44)(25,39)(27,55,56,53)(31,51,61,57)(32,63,59,50)(33,54)(46,48,58,62);(1,4)(2,3)(5,22)(6,23)(7,37)(8,25)(9,26)(10,42)(11,17)(12,18)(13,34)(14,20)(15,21)(16,45)(19,33)(24,38)(27,50)(28,44)(29,41)(30,52)(31,62)(32,53)(35,47)(36,54)(39,40)(43,49)(46,57)(48,61)(51,58)(55,59)(56,63)(60,64);(1,5,7,31,8,32,36,60,9,30,38,61,35,59,6,29)(2,10,13,27,14,46,11,43,15,45,33,56,39,58,12,44)(3,16,34,63,20,51,17,28,21,42,19,50,40,57,18,49)(4,22,37,62,25,53,54,64,26,52,24,48,47,55,23,41)

Polycyclic presentation:

2^10|a^2=g,b^2=g,d^2=fi,e^2=ij,f^2=ij,h^2=i,i^2=j,c^b=cg,d^a=dh,d^b=de,d^c=defi,e^a=ei,e^b=eij,f^a=fij,f^b=fi,g^d=gj,h^a=hi,h^b=hi,h^c=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307537
maximal normal subgroup of	#72307537

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