#510520

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

431 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)(3,4)(5,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,31)(21,24)(22,27)(23,26)(28,41)(29,42)(30,37)(32,44)(33,38)(34,43)(35,45)(36,40)(39,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(18,24)(19,28)(20,23)(22,34)(25,39)(27,44)(29,47)(30,46)(32,33)(35,36)(37,41)(38,43)(40,48)(42,45);(1,4,5,6)(2,3,7,8)(10,14)(13,16)(17,19,21,30)(18,25,24,37)(20,29,23,36)(22,35,33,47)(26,40,31,42)(27,45,38,48)(28,32,46,34)(39,43,41,44);(2,7)(3,8)(10,14)(13,16)(17,20,21,23)(18,26,24,31)(19,29,30,36)(22,32,33,34)(25,40,37,42)(27,43,38,44)(28,35,46,47)(39,45,41,48)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,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, 417716);