#6351194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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

2600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258390);