#6349720

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
prime radical	#62831	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2625 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^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, 6256916);