#6349285

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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

2802 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)(8,14)(10,16)(13,15)(17,18)(19,24)(20,32)(21,27)(22,26)(23,28)(25,33)(29,43)(30,39)(31,40)(34,41)(35,38)(36,42)(37,46)(44,48)(45,47);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,37)(26,38)(27,39)(28,40)(32,47)(33,44)(34,45)(36,46)(41,43)(42,48);(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,32)(27,41)(28,42)(30,44)(31,45)(35,46)(38,47)(39,43)(40,48);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,40,39)(25,32,42,41)(29,44,45,46)(37,47,48,43);(3,10)(6,15)(9,16)(11,13)(17,21,23,22)(18,27,28,26)(19,30,31,35)(20,33,34,36)(24,39,40,38)(25,41,42,32)(29,44,45,46)(37,43,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256481);