#6349732

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2887 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,15)(14,16)(17,18)(19,27)(20,24)(21,33)(22,25)(23,30)(26,35)(28,41)(29,40)(31,38)(32,45)(34,37)(36,43)(39,44)(42,48)(46,47);(2,7)(4,5)(8,15)(9,10)(11,16)(13,14)(17,19)(18,23)(20,28)(21,34)(22,29)(24,36)(25,37)(26,40)(27,33)(30,35)(31,42)(32,46)(38,47)(39,48)(41,45)(43,44);(1,3)(2,8)(4,13)(5,14)(6,12)(7,15)(9,11)(10,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,38)(26,39)(27,41)(29,42)(30,43)(33,45)(34,46)(35,44)(37,47)(40,48);(3,12)(8,16)(11,15)(13,14)(18,25)(20,31)(23,37)(26,33)(27,40)(28,42)(32,44)(43,46);(1,4,6,5)(2,9,10,7)(3,13,12,14)(8,11,16,15)(18,25)(19,29)(24,38)(26,33)(28,42)(30,34)(39,45)(43,46)

Polycyclic presentation:

2^9|e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6256928);