Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,6)(4,10,14,16)(11,13,12,15)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(34,39)(35,42)(36,44)(46,47);(1,3)(2,8)(4,12)(5,7)(6,9)(10,15)(11,14)(13,16)(17,19)(18,24)(20,27)(21,22)(23,26)(25,31)(28,30)(29,32)(33,34)(35,37)(36,45)(38,39)(40,44)(41,42)(43,46)(47,48);(2,9)(3,5)(10,16)(11,12)(19,26)(21,24)(27,32)(28,31)(33,35,38,42)(34,37,39,41)(36,43,44,48)(40,47,45,46);(1,4)(2,10)(3,11)(5,12)(6,15)(7,14)(8,13)(9,16)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,36)(34,40)(35,43)(37,47)(38,44)(39,45)(41,46)(42,48);(1,5,7,3)(2,8,9,6)(4,12,14,11)(10,13,16,15)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,35)(34,41)(36,43)(37,39)(38,42)(40,46)(44,48)(45,47);(1,6,7,8)(2,3,9,5)(4,13,14,15)(10,12,16,11)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,37,38,41)(34,35,39,42)(36,46,44,47)(40,48,45,43)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=gh,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=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541699);