#387064

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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)(11,15)(12,14)(13,16)(17,18,22,26,21,24,19,23)(20,28,27,32,29,25,30,31);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,22)(18,23,24,26)(20,27,29,30)(25,31,28,32);(1,4,5,8)(2,6,3,7)(11,16)(13,15)(18,24)(19,22)(25,28)(27,30);(2,3)(6,7)(10,14)(11,16)(17,20,21,29)(18,25,24,28)(19,27,22,30)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=egi,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294260);