#6481618

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2865 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,22,25)(19,27,29,43)(20,24,33,41)(21,34,37,26)(23,35,39,30)(28,46,31,45)(32,44,36,42)(38,48,40,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,23)(20,31)(21,35)(22,29)(24,40)(25,39)(26,43)(27,34)(28,33)(30,37)(32,47)(36,48)(38,41)(42,46)(44,45);(3,9)(6,13)(10,11)(14,15)(17,20,22,33)(18,24,25,41)(19,28,29,31)(21,36,37,32)(23,38,39,40)(26,44,34,42)(27,45,43,46)(30,48,35,47);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(18,25)(19,29)(24,41)(26,34)(28,31)(30,35)(42,44)(47,48);(2,8)(6,13)(7,16)(14,15)(18,25)(23,39)(24,41)(26,34)(27,43)(38,40)(42,44)(45,46)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=hi,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, 6388814);