#493890

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fi,g^c=ghi,g^f=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, 401086);