#387238

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=dei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294434);