#6349307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349298
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

2786 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)(19,28)(20,25)(21,26)(22,36)(23,27)(24,30)(29,43)(31,39)(32,38)(33,40)(34,41)(35,42)(37,45)(44,47)(46,48);(4,13)(8,12)(9,16)(11,14)(17,19,23,32)(18,24,27,39)(20,29,35,46)(21,30,22,31)(25,37,42,47)(26,38,36,28)(33,44,34,45)(40,43,41,48);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,37)(26,41)(27,42)(28,43)(30,45)(31,44)(32,46)(36,40)(38,48)(39,47);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,43,47,48);(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,33)(24,28,39,38)(25,41,42,40)(29,44,46,45)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256503);