#425643

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1074 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fi,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332839);