#6489609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

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

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, 6396805);