#423825

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1348 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)(9,10,14,19,15,20,30,32)(11,21,23,18,24,37,13,16)(12,25,27,36,28,17,29,35)(22,33,39,31,40,38,26,34);(9,11,15,24)(10,16,20,18)(12,26,28,39)(13,30,23,14)(17,33,25,38)(19,37,32,21)(22,27,40,29)(31,35,34,36);(1,2,3,4)(9,12,15,28)(10,17,20,25)(11,22,24,40)(13,26,23,39)(14,27,30,29)(16,31,18,34)(19,35,32,36)(21,38,37,33);(6,8)(9,13,14,11,15,23,30,24)(10,18,19,37,20,16,32,21)(12,26,27,22,28,39,29,40)(17,34,35,33,25,31,36,38)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 331021);