#429641

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336837);