#6351710

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258906);