#6468041

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di

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, 6375237);