#415058

order	512
nilpotency class	3
isoclinism	isoclinic class of #395552
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1096 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)(3,7)(4,5)(6,8)(9,10,13,29,15,18,14,19)(11,20,22,16,24,33,23,32)(12,25,30,17,28,36,27,35)(21,31,26,34,38,39,37,40);(9,11,13,22,15,24,14,23)(10,16,19,20,18,32,29,33)(12,26,27,21,28,37,30,38)(17,34,36,39,35,40,25,31);(2,5)(6,7)(9,12)(10,17)(11,21)(13,27)(14,30)(15,28)(16,31)(18,35)(19,36)(20,34)(22,37)(23,26)(24,38)(25,29)(32,39)(33,40);(1,3,4,8)(2,6,5,7)(10,18)(16,32)(17,35)(19,29)(20,33)(25,36)(31,39)(34,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,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, 322254);