#451284

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

2^9|a^2=efghi,b^2=ef,c^2=i,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^d=hi

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, 358480);