#496339

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1125 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)(4,6)(5,8)(9,10)(11,19)(12,23)(13,18)(14,27,26,16)(15,20)(17,24)(21,29)(22,30,35,28)(25,31,37,32)(33,34)(36,38,40,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,30)(17,29)(18,19)(23,33)(24,34)(25,36)(26,35)(27,28)(31,39)(32,38)(37,40);(3,8)(5,7)(10,16)(11,20)(12,24)(15,28)(17,32)(18,27)(19,30)(22,35)(23,31)(25,37)(29,39)(33,38);(2,6)(5,7)(9,12,13,24)(10,17,18,23)(11,21,20,34)(14,25,26,37)(15,29,19,33)(16,31,27,32)(22,36,35,40)(28,38,30,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403535);