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 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,6)(5,12)(7,11)(8,14)(10,15)(13,16)(18,23)(19,25)(21,29)(28,31)(33,34,39,41)(35,37,43,36)(38,40,47,46)(42,45,48,44);(2,4)(7,9)(8,12)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,39,43)(34,36,41,37)(38,42,47,48)(40,44,46,45);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36,39,37)(34,35,41,43)(38,44,47,45)(40,42,46,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,34)(35,36)(37,43)(38,46)(39,41)(40,47)(42,45)(44,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37,39,36)(34,35,41,43)(38,45,47,44)(40,42,46,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,38,39,47)(34,40,41,46)(35,42,43,48)(36,44,37,45)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10315560);