#6351683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351670
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	#62825	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2664 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6258879);