#391155

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1236 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)(3,5)(4,7)(6,8)(10,13)(11,15)(17,18)(19,21)(20,22)(23,24)(25,26,30,28,29,32,27,31);(1,3)(2,4)(5,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,30)(26,31,32,28);(1,4,6,5)(2,7,8,3)(9,11,12,15)(10,14,13,16)(18,22)(19,23)(25,28,30,32,29,31,27,26);(1,5,6,4)(2,7,8,3)(10,13)(14,16)(18,22)(21,24)(25,27,29,30)(26,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=eh,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de

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, 298351);