#6350099

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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

2776 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,23)(20,30)(21,33)(22,25)(24,32)(26,36)(27,43)(28,41)(29,38)(31,42)(34,39)(35,40)(37,45)(44,47)(46,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,22,29)(18,23,25,38)(20,27,31,44)(21,34,36,28)(24,37,40,48)(26,41,33,39)(30,47,42,43)(32,46,35,45);(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,40)(26,42)(28,46)(29,44)(30,33)(32,36)(34,45)(38,48)(39,47)(41,43);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(18,25)(23,38)(24,40)(26,33)(30,42)(37,48)(39,41)(43,47);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)

Polycyclic presentation:

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

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, 6257295);