#483088

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

767 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)(17,18,22,28,37,44,35,45,23,25,21,27,20,26,36,46)(19,29,32,43,30,48,38,24,34,42,31,41,33,47,39,40);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,24)(20,30)(21,38)(22,39)(23,34)(25,40)(26,41)(27,47)(28,48)(29,45)(31,35)(32,36)(33,37)(42,46)(43,44);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,25)(24,40)(26,44)(27,28)(29,42)(41,43)(45,46)(47,48);(17,20,23,37)(18,26,25,44)(19,30,34,33)(21,35,22,36)(24,41,40,43)(27,45,28,46)(29,48,42,47)(31,39,32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390284);