#403865

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=gi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311061);