#359267

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266463);