#6477403

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2963 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,44)(31,38)(32,40)(33,41)(34,42)(36,39)(37,45)(43,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,45)(33,46)(34,47)(40,43)(41,44)(42,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,34)(24,37)(26,41)(27,42)(28,43)(30,46)(31,47)(35,40)(36,45)(38,44)(39,48);(3,10)(6,14)(9,11)(13,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,34,33)(24,38,39,28)(25,40,42,41)(29,45,47,46)(37,43,48,44);(2,8)(6,14)(7,16)(13,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,34,32)(24,38,39,28)(25,41,42,40)(29,46,47,45)(37,44,48,43)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,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, 6384599);