#409923

order	512
nilpotency class	3
isoclinism	isoclinic class of #395154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fi,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di

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, 317119);