#6592423

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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)(20,28)(21,26)(22,23)(24,27)(29,31)(30,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,28,32,31);(11,15)(13,16)(17,20,21,29)(18,24,26,32)(19,27,22,30)(23,28,25,31);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,28)(29,32)(30,31);(11,15)(13,16)(17,19,21,22)(18,25,26,23)(20,27,29,30)(24,31,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6499619);