#6489631

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470058
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	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2780 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,23,26)(19,29,31,43)(20,33,36,44)(21,28,22,27)(24,37,40,32)(25,38,42,35)(30,39,34,41)(45,46,48,47);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,19,23,31)(18,24,26,40)(20,34,36,30)(21,37,22,32)(25,41,42,39)(27,43,28,29)(33,45,44,48)(35,46,38,47);(1,4,6,5)(2,8,10,9)(3,11,13,12)(7,14,16,15)(17,20,23,36)(18,25,26,42)(19,30,31,34)(21,35,22,38)(24,39,40,41)(27,33,28,44)(29,45,43,48)(32,46,37,47);(1,5,6,4)(2,9,10,8)(3,12,13,11)(7,15,16,14)(18,26)(19,31)(25,42)(27,28)(30,34)(32,37)(33,44)(46,47);(2,10)(7,16)(8,9)(14,15)(18,26)(24,40)(25,42)(27,28)(29,43)(33,44)(39,41)(45,48)

Polycyclic presentation:

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

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, 6396827);