#495399

order	512
nilpotency class	4
isoclinism	isoclinic class of #480454
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fi,d^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=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, 402595);