#495430

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

949 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,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, 402626);