#422493

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1287 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329689);