#6351870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351869
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

2821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,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, 6259066);