#496220

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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1672 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)(9,10,13,18,14,19,27,37)(11,15,22,31,20,29,38,40)(12,17,24,34,25,28,26,36)(16,32,33,21,30,39,35,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,28)(17,30)(18,31)(19,29)(23,25)(24,39)(26,32)(27,38)(33,36)(34,35)(37,40);(3,8)(5,7)(9,11)(10,16)(12,23)(13,22)(14,20)(15,28)(17,29)(18,35)(19,30)(21,25)(24,32)(26,39)(27,38)(31,34)(33,37)(36,40);(2,6)(5,7)(11,20)(15,29)(16,30)(21,23)(22,38)(31,40)(32,39)(33,35)

Polycyclic presentation:

2^9|a^2=g,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,d^c=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, 403416);