#129072

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₄⋊C₄
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,3,4,7,6,8)(9,10,14,19,15,20,32,29,16,21,33,28,13,18,27,12)(11,22,25,23,24,36,39,38,26,37,31,40,30,17,34,35);(1,3,5,7,4,8,6,2)(9,11)(10,17)(12,23)(13,30)(14,31)(15,24)(16,26)(18,37)(19,38)(20,22)(21,36)(25,33)(27,39)(28,35)(29,40)(32,34);(1,2,6,8,4,7,5,3)(9,12,27,18,13,28,33,21,16,29,32,20,15,19,14,10)(11,23,39,37,30,35,25,36,26,40,34,22,24,38,31,17)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ch,d^b=di,e^b=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36268);