#507312

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,30)(20,26)(21,37)(22,28)(23,29)(24,27)(25,33)(31,39)(32,40)(34,45)(35,44)(36,43)(38,42)(41,46)(47,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,31,46,21,20,35,47,22,33,23,34,24,32,38,36)(18,25,39,45,37,40,44,43,28,30,29,41,27,26,42,48);(10,14)(13,15)(17,20,22,32)(18,26,28,40)(19,24,33,21)(23,34,31,46)(25,37,30,27)(29,45,39,41)(35,36,38,47)(42,48,44,43);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21,22,24)(18,27,28,37)(19,32,33,20)(23,38,31,35)(25,40,30,26)(29,44,39,42)(34,47,46,36)(41,48,45,43)

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414508);