#379036

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=f,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,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, 286232);