#72260471

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×#2263	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

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=fh,c^2=efgj,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,e^c=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, #72322349, #72322516
maximal normal subgroup of	#72260475, #72322349, #72322516

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.6*F.7*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, 5, 3, 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);