#391004

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

892 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)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(25,26,30,28,29,32,27,31);(1,2,4,5)(3,8,7,6)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)(25,27,29,30)(26,28,32,31);(9,12,13,11)(10,14,15,16)(18,22)(19,23)(25,28,30,32,29,31,27,26);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(18,22)(21,24)(25,27,29,30)(26,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=fgh,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, 298200);