#495644

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×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1049 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)(11,19,22,33)(12,23)(13,17,27,31)(14,18)(15,21)(16,24)(20,34,35,32)(25,38,39,30)(26,29)(28,40)(36,37);(9,11,13,26)(10,15,17,19)(12,20,25,40)(14,22,27,21)(16,28,30,32)(18,29,31,33)(23,37,38,34)(24,35,39,36);(2,5)(6,8)(9,12)(10,16)(11,20)(13,25)(14,24)(15,28)(17,30)(18,23)(19,32)(21,36)(22,35)(26,40)(27,39)(29,37)(31,38)(33,34);(1,3,4,7)(2,6,5,8)(12,24)(16,23)(20,35)(25,39)(28,37)(30,38)(32,34)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=fi,e^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, 402840);