#7195131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(6,8)(10,13)(11,15)(17,18,23,27)(19,29,31,20)(21,25,30,32)(22,28,24,26);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,23,31)(18,20,27,29)(21,24,30,22)(25,28,32,26);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,25,23,29,24,32)(18,21,26,31,27,30,28,19);(2,5)(6,8)(10,13)(14,16)(17,21,23,30)(18,25,27,32)(19,22,31,24)(20,26,29,28);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,26,27,28)(19,30,31,21)(20,32,29,25)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fghi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102327);