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 | |
4298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(33,34)(35,40)(36,42)(37,43)(38,45)(39,41)(44,48)(46,47);(2,6)(4,12)(7,10)(11,15)(17,19,22,30)(18,23,24,27)(20,28,31,26)(21,25,32,29)(33,35,39,37)(34,40,41,43)(36,45,46,44)(38,47,48,42);(3,9)(7,10)(11,15)(14,16)(18,24)(20,31)(23,27)(26,28)(34,41)(35,37)(42,47)(44,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36)(34,42)(35,44)(37,45)(38,43)(39,46)(40,48)(41,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,37,39,35)(34,43,41,40)(36,45,46,44)(38,47,48,42);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,38)(34,44)(35,42)(36,43)(37,47)(39,48)(40,46)(41,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232818);