#412062

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319258);