#6352003

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

2626 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=i,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, 6259199);