#6351535

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351503
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	#60702	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2608 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,33)(21,26)(22,28)(23,27)(24,31)(25,34)(30,38)(32,39)(35,41)(36,40)(37,42)(43,45)(44,47)(46,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,40)(28,39)(33,47)(34,45)(35,46)(37,44)(41,43)(42,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20,23,35)(18,25,27,42)(19,30,32,46)(21,34,22,37)(24,38,40,48)(26,41,28,33)(29,43,39,47)(31,44,36,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,40,29)(25,41,42,33)(30,44,46,45)(38,47,48,43);(18,27)(24,40)(25,42)(26,28)(29,39)(33,41)(38,48)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258731);