#495254

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1354 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)(6,8)(9,10,15,20,29,18,28,37,14,17,13,19,12,21,30,36)(11,22,26,34,23,39,31,16,25,35,24,40,27,38,32,33);(1,2,3,4)(9,11,14,25)(10,16,17,33)(12,23,29,27)(13,31,15,32)(18,34,21,40)(19,38,20,39)(22,36,35,37)(24,30,26,28);(5,6,7,8)(10,17)(16,33)(18,21)(19,20)(22,35)(34,40)(36,37)(38,39);(9,12,14,29)(10,18,17,21)(11,23,25,27)(13,28,15,30)(16,34,33,40)(19,36,20,37)(22,38,35,39)(24,32,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,d^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, 402450);