#494444

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1496 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,h^b=hi

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, 401640);