#6592430

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(18,22)(19,25)(21,28)(23,31)(26,32)(27,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,22)(21,24)(23,27)(25,31)(28,30)(29,32);(11,15)(13,16)(17,19,20,27)(18,23,22,26)(21,29,28,25)(24,31,30,32);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(19,27)(23,26)(25,29)(31,32);(11,15)(13,16)(18,22)(23,26)(24,30)(31,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=fi,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, 6499626);