#6351979

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2639 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:

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

Polycyclic presentation:

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

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, 6259175);