#493860

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1683 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)(11,15)(12,17)(13,25,26,16)(14,18)(19,28)(20,34,35,27)(21,29)(22,31)(23,37,38,30)(24,32)(33,40,36,39);(2,5)(6,8)(9,11,14,21)(10,15,18,29)(12,22,24,19)(13,20,26,35)(16,27,25,34)(17,31,32,28)(23,36,38,33)(30,40,37,39);(1,3,4,7)(2,6,5,8)(10,16)(15,27)(17,30)(18,25)(28,39)(29,34)(31,40)(32,37);(9,12,14,24)(10,17,18,32)(11,19,21,22)(13,23,26,38)(15,28,29,31)(16,30,25,37)(20,33,35,36)(27,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,g^c=gh,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, 401056);