#480239

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387435);