#495650

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

963 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)(3,6)(4,5)(7,8)(9,10,15,21,16,22,12,18)(11,23,25,38,27,34,30,40)(13,19,28,37,14,20,24,32)(17,31,35,29,36,39,33,26);(9,11,24,31,16,27,28,39)(10,17,32,38,22,36,37,40)(12,25,13,26,15,30,14,29)(18,33,19,34,21,35,20,23);(2,5)(6,8)(9,12,16,15)(10,18,22,21)(11,25,27,30)(13,28,14,24)(17,33,36,35)(19,37,20,32)(23,38,34,40)(26,39,29,31);(1,3,4,7)(2,6,5,8)(9,13,16,14)(10,19,22,20)(11,26,27,29)(12,24,15,28)(17,34,36,23)(18,32,21,37)(25,31,30,39)(33,38,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=fi,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, 402846);