#495570

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	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 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)(9,10,13,18,15,20,33,21)(11,22,27,39,26,37,25,30)(12,28,16,35,31,17,29,36)(14,34,32,19,23,24,40,38);(9,11,23,17,13,25,40,35,15,26,14,28,33,27,32,36)(10,16,19,37,21,31,34,30,20,29,38,22,18,12,24,39);(2,5)(6,8)(9,12)(10,17)(11,24)(13,16)(14,30)(15,31)(18,36)(19,25)(20,28)(21,35)(22,32)(23,39)(26,34)(27,38)(29,33)(37,40);(1,3,4,7)(2,6,5,8)(9,13,15,33)(10,18,20,21)(11,25,26,27)(12,29,31,16)(14,32,23,40)(17,35,28,36)(19,24,38,34)(22,30,37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,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, 402766);