#434248

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341444);