#480165

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,f^b=fg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387361);