#452512

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1200 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)(11,15,16,13)(17,18,20,22)(19,21,23,24)(25,26,30,33,28,32,29,34)(27,35,38,39,36,31,37,40);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,23)(22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,2,5,4)(3,6,8,7)(10,14)(13,15)(19,23)(21,24)(26,32)(31,35)(33,34)(39,40);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(26,33)(27,36)(29,30)(31,39)(32,34)(35,40)

Polycyclic presentation:

2^9|a^2=egh,b^2=fi,c^2=ef,d^2=ef,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, 359708);