#466827

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1412 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:

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

Polycyclic presentation:

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

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, 374023);