#495390

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
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×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

842 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)(3,7)(4,5)(6,8)(9,10,15,19,16,21,14,20)(11,22,24,13,25,31,18,32)(12,26,30,33,27,35,29,17)(23,37,39,36,38,28,34,40);(2,5)(6,7)(9,11,13,21,15,18,31,19,16,25,32,10,14,24,22,20)(12,23,28,33,30,34,40,26,27,38,37,17,29,39,36,35);(1,3,4,8)(2,6,5,7)(9,12,14,29,16,27,15,30)(10,17,19,26,21,33,20,35)(11,23,24,39,25,38,18,34)(13,28,22,36,32,37,31,40);(12,27)(17,33)(23,38)(26,35)(28,37)(29,30)(34,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402586);