#72260464

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

617 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=efghij,e^2=gi,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, #72322334, #72322516
maximal normal subgroup of	#72260475, #72322334, #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.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.7*F.9);;
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);