#7199296

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1578 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)(6,7)(10,14)(12,15)(19,25)(21,26)(23,30)(24,29)(27,31)(28,32);(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,12,16,15)(17,18,22,20)(19,23,27,28)(21,24,26,29)(25,30,31,32);(9,11)(10,15)(12,14)(13,16)(17,19,26,31,22,27,21,25)(18,23,29,32,20,28,24,30);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20,22,18)(19,23,27,28)(21,29,26,24)(25,30,31,32);(9,12,13,15)(10,11,14,16)(18,20)(23,28)(24,29)(30,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=fhi,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=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106492);