#472830

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(9,11,13,16)(10,14,12,15)(17,19,20,23)(18,21,22,24)(25,27,29,35,28,34,36,33)(26,30,32,37,31,38,39,40);(2,5)(6,8)(10,12)(14,15)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(18,22)(21,24)

Polycyclic presentation:

2^9|a^2=e,b^2=egh,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380026);