#495624

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1016 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)(11,19,23,33)(12,24)(13,27,20,17)(14,18)(15,22)(16,25)(21,35,36,32)(26,31,34,39)(28,30)(29,40)(37,38);(9,11,20,22,14,23,13,28)(10,15,27,33,18,30,17,19)(12,21,34,37,25,36,26,40)(16,29,39,35,24,38,31,32);(2,5)(6,8)(9,12)(10,16)(11,21)(13,26)(14,25)(15,29)(17,31)(18,24)(19,32)(20,34)(22,37)(23,36)(27,39)(28,40)(30,38)(33,35);(1,3,4,7)(2,6,5,8)(12,25)(16,24)(21,36)(26,34)(29,38)(31,39)(32,35)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fi,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,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, 402820);