#126811

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

457 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,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, 34007);