#6477398

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	C₂×#3493	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2954 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)(17,18,22,26)(19,27,30,41)(20,24,31,39)(21,33,36,25)(23,34,38,29)(28,44,45,43)(32,42,35,40)(37,47,48,46);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,41)(26,38)(27,33)(29,36)(31,45)(32,47)(35,46)(39,48)(40,43)(42,44);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20,22,31)(18,24,26,39)(19,28,30,45)(21,35,36,32)(23,37,38,48)(25,42,33,40)(27,43,41,44)(29,47,34,46);(3,10)(6,14)(9,11)(13,15)(20,31)(24,39)(28,45)(32,35)(37,48)(40,42)(43,44)(46,47);(2,8)(6,14)(7,16)(13,15)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,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, 6384594);