#6349297

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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
prime radical	#62788	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2759 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256493);