#72260469

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2177
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#2262	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ #11399 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,5,7,4,6,8,3)(11,21,24,49)(12,27)(13,32)(14,33)(16,35,38,64)(18,43)(19,46)(20,42)(22,47,40,53)(23,30)(25,59,58,56)(26,61,36,51)(28,52)(29,34)(31,63)(37,45)(39,69,68,67)(41,44)(48,71,60,50)(54,72,70,66)(55,65)(57,62);(1,3,8,6,4,7,5,2)(9,10)(11,22,24,40)(12,28)(13,19)(14,20)(15,17)(16,36,38,26)(18,31)(21,47,49,53)(23,55)(25,54,58,70)(27,52)(29,44)(30,65)(32,46)(33,42)(34,41)(35,51,64,61)(37,57)(39,48,68,60)(43,63)(45,62)(50,69,71,67)(56,66,59,72);(2,6)(3,7)(9,11,23,56,13,25,31,49,33,36,63,71,29,48,30,51,15,24,57,59,34,58,12,21,14,26,27,50,32,60,62,61)(10,16,37,67,19,39,28,64,42,22,52,72,44,54,45,47,17,38,55,69,41,68,18,35,20,40,43,66,46,70,65,53);(10,17)(16,38)(18,28)(19,41)(20,42)(22,40)(35,64)(37,55)(39,68)(43,52)(44,46)(45,65)(47,53)(54,70)(66,72)(67,69)

Polycyclic presentation:

2^10|a^2=f,b^2=fh,c^2=eghij,e^2=gij,f^2=h,g^2=ij,i^2=j,c^a=ce,c^b=chij,d^b=dj,e^a=eg,e^b=ej,f^c=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260475, #72322316, #72322555
maximal normal subgroup of	#72260475, #72322316, #72322555

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