#391168

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,21)(19,23)(25,26,30,28,29,32,27,31);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,18,20,21)(19,24,23,22)(25,27,29,30)(26,31,32,28);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(18,21)(22,24)(25,28,30,32,29,31,27,26);(2,3)(6,7)(10,14)(13,16)(17,19,20,23)(18,22,21,24)(26,32)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=egh,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,d^c=di

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, 298364);