#494114

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,27,33,41,20,25,31,39,32,40,44,48)(19,23,29,34,30,35,43,45,28,24,36,37,42,38,46,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,28)(18,23,25,24)(21,29,31,36)(22,30,32,42)(26,34,39,37)(27,35,40,38)(33,43,44,46)(41,45,48,47);(10,14)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,25)(26,37)(27,38)(31,36)(32,42)(33,43)(34,39)(35,40)(41,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(19,28)(23,24)(29,36)(30,42)(34,37)(35,38)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401310);