#405074

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, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=g,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, 312270);