#413091

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=e,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=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, 320287);