#7623945

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531141);