#6592527

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1679 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,26)(21,23)(22,30)(24,28)(27,32)(29,31);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(18,23)(19,27)(26,30)(28,29);(11,15)(13,16)(17,19,21,27)(18,24,23,31)(20,28,22,29)(25,26,32,30);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,29)(21,27)(22,28)(23,32)(24,30)(26,31);(11,15)(13,16)(17,20,21,22)(18,26,23,30)(19,28,27,29)(24,32,31,25)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6499723);