#7624207

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531403);