#8639923

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8547119);