#496217

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1648 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)(3,7)(4,6)(5,8)(9,10,14,19,15,20,12,25)(11,16,24,32,22,30,21,35)(13,18,26,33,28,34,27,17)(23,31,36,39,37,40,38,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,29)(18,31)(19,32)(20,30)(25,35)(26,36)(27,38)(28,37)(33,39)(34,40);(3,8)(5,7)(9,12,15,14)(10,17,20,33)(11,21,22,24)(13,27,28,26)(16,29,30,39)(18,19,34,25)(23,38,37,36)(31,32,40,35);(2,6)(5,7)(11,22)(16,30)(21,24)(23,37)(29,39)(31,40)(32,35)(36,38)

Polycyclic presentation:

2^9|a^2=g,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^c=fi,g^c=ghi,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, 403413);