#6626998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,18,22,24)(19,21,26,23)(20,25,29,31)(27,30,32,28);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(18,24)(19,26)(25,31)(27,32);(9,11,13,12)(10,15,14,16)(17,19,23,18,22,26,21,24)(20,27,28,25,29,32,30,31);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,20)(18,25)(19,27)(21,30)(22,29)(23,28)(24,31)(26,32);(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,24,19)(20,28,29,30)(25,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6534194);