#483231

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=gi,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

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, 390427);