#387713

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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

1168 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)(12,16)(13,14)(17,18,20,24,21,23,27,30)(19,22,25,29,26,28,31,32);(1,3,5,2)(4,7,8,6)(10,14)(11,12)(17,19,21,26)(18,22,23,28)(20,25,27,31)(24,29,30,32);(1,4,5,8)(2,6,3,7)(9,11,13,12)(10,15,14,16)(18,23)(20,27)(22,28)(25,31);(2,3)(6,7)(9,12,13,11)(10,15,14,16)(18,23)(20,27)(22,28)(25,31)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=e,d^2=g,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, 294909);