#6351825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

2769 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)(21,27)(22,26)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,42)(37,41)(43,45)(44,47)(46,48);(4,13)(8,12)(9,16)(11,14)(17,19,23,33)(18,24,28,40)(20,30,37,46)(21,31,22,32)(25,38,42,48)(26,39,27,29)(34,47,41,43)(35,44,36,45);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,34)(27,41)(28,42)(29,43)(31,45)(32,44)(33,46)(39,47)(40,48);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,40,29)(25,41,42,34)(30,44,46,45)(38,43,48,47);(17,21,23,22)(18,27,28,26)(19,32,33,31)(20,36,37,35)(24,39,40,29)(25,41,42,34)(30,44,46,45)(38,47,48,43)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6259021);