#126823

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cgi,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34019);