#72260461

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	C₂²×D₃₂	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

633 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)(12,23)(13,28)(14,31)(16,33)(18,35)(19,41)(20,40)(22,43)(24,45)(25,56)(26,57)(27,51)(29,62)(30,54)(32,61)(34,47)(36,58)(37,68)(38,49)(39,59)(42,48)(44,46)(50,71)(52,53)(55,72)(60,66)(63,64)(65,69)(67,70);(1,3,4,7,5,8,2,6)(9,10)(11,22,24,38)(12,27)(13,19)(14,20)(15,17)(16,34,36,26)(18,30)(21,43,45,49)(23,51)(25,50,55,63)(28,41)(29,60)(31,40)(32,39)(33,47,58,57)(35,54)(37,65,67,44)(42,52)(46,68,69,70)(48,53)(56,71,72,64)(59,61)(62,66);(1,2,5,4)(3,7,8,6)(9,11,23,46,13,25,52,57,31,34,62,72,61,65,30,45,15,24,54,69,32,55,29,47,14,26,53,56,28,44,12,21)(10,16,35,64,19,37,60,49,40,22,48,70,59,50,27,58,17,36,51,71,39,67,42,43,20,38,66,68,41,63,18,33);(10,17)(16,36)(18,27)(19,39)(20,40)(22,38)(33,58)(35,51)(37,67)(41,59)(42,60)(43,49)(48,66)(50,63)(64,71)(68,70)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=egh,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,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, #72322501
maximal normal subgroup of	#72260475, #72322316, #72322501

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.7*F.8);;
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, 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);