#7195518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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
prime radical	#45388	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=f,e^2=hi,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^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, 7102714);