#375917

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1122 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)(3,7)(4,6)(5,8)(9,10,16,21,14,20,15,36)(11,22,26,28,24,17,25,37)(12,27,30,18,29,39,31,38)(13,32,35,23,33,19,34,40);(1,3,4,8)(2,5,6,7)(9,11,14,24)(10,17,20,22)(12,19,29,32)(13,18,33,38)(15,25,16,26)(21,37,36,28)(23,30,40,31)(27,35,39,34);(9,12)(10,18)(11,23)(13,28)(14,29)(15,31)(16,30)(17,35)(19,26)(20,38)(21,39)(22,34)(24,40)(25,32)(27,36)(33,37);(2,6)(5,7)(9,13,16,35,14,33,15,34)(10,19,21,40,20,32,36,23)(11,18,26,39,24,38,25,27)(12,28,30,17,29,37,31,22)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283113);