#495645

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

980 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)(3,6)(4,5)(7,8)(9,10)(11,19,22,33)(12,16)(13,17,26,31)(14,18)(15,21)(20,32,34,40)(23,29)(24,30,36,39)(25,28)(27,35)(37,38);(9,11,13,25)(10,15,17,19)(12,20,24,37)(14,22,26,21)(16,27,30,32)(18,28,31,33)(23,34,36,35)(29,38,39,40);(2,5)(6,8)(9,12,14,23)(10,16,18,29)(11,20,22,34)(13,24,26,36)(15,27,28,38)(17,30,31,39)(19,32,33,40)(21,35,25,37);(1,3,4,7)(2,6,5,8)(12,23)(16,29)(20,34)(24,36)(27,38)(30,39)(32,40)(35,37)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=f,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, 402841);