#496248

order	512
nilpotency class	4
isoclinism	isoclinic class of #480670
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

1658 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23)(13,18)(14,28,29,19)(15,20)(17,25,31,22)(21,30)(24,33)(26,39,40,35)(27,36)(32,38,34,37);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,16,20,23)(13,24,27,21)(14,22,29,25)(17,19,31,28)(18,33,36,30)(26,38,40,37)(32,39,34,35);(3,8)(5,7)(9,12,15,11)(10,17,20,31)(13,21,27,24)(14,25,29,22)(16,19,23,28)(18,34,36,32)(26,37,40,38)(30,39,33,35);(2,6)(5,7)(9,13,15,27)(10,18,20,36)(11,21,12,24)(14,26,29,40)(16,30,23,33)(17,32,31,34)(19,35,28,39)(22,37,25,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403444);