#402204

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=e,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^c=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, 309400);