#6351398

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2668 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)(20,34)(21,27)(22,28)(23,26)(24,32)(25,35)(30,38)(31,39)(33,42)(36,41)(37,40)(43,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,39)(27,29)(28,42)(34,47)(35,44)(36,46)(37,45)(40,48)(41,43);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,40)(27,41)(28,34)(29,43)(31,44)(32,45)(33,46)(39,48)(42,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,26)(24,39)(25,40)(27,28)(29,42)(34,41)(38,48)(43,47);(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,35,36,37)(24,29,39,42)(25,41,40,34)(30,44,46,45)(38,43,48,47)

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=ghi,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, 6258594);