#9017526

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

4259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924722);