#495619

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
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×Q₈	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

955 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)(9,10,14,20,15,18,12,17)(11,21,23,37,25,38,29,40)(13,19,26,35,28,36,22,31)(16,30,34,27,33,39,32,24);(9,11,22,39,15,25,26,30)(10,16,31,40,18,33,35,37)(12,23,13,27,14,29,28,24)(17,32,19,21,20,34,36,38);(2,5)(6,8)(9,12,15,14)(10,17,18,20)(11,23,25,29)(13,26,28,22)(16,32,33,34)(19,35,36,31)(21,37,38,40)(24,39,27,30);(1,3,4,7)(2,6,5,8)(10,18)(16,33)(17,20)(19,36)(21,38)(31,35)(32,34)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,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, 402815);