#413433

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320629);