#410636

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fgi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei

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, 317832);