#416350

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1258 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)(9,10,13,14)(11,15,12,16)(17,18)(19,22)(20,28)(21,26)(23,25)(24,30)(27,31)(29,32);(5,6)(7,8)(10,14)(15,16)(17,19,22,18,23,26,21,25)(20,28,30,32,31,27,29,24);(2,4)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,29,23,31,22,30)(18,24,19,27,25,32,26,28);(9,12,13,11)(10,15,14,16)(18,25)(19,26)(24,32)(27,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 323546);