#420537

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

1679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=e,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,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, 327733);