#483095

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
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	#60714	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

771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390291);