#483262

order	512
nilpotency class	4
isoclinism	isoclinic class of #482975
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	#62418	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

727 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390458);