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 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(18,22)(19,26)(20,28)(30,31)(33,34)(35,40)(36,37)(38,43)(39,46)(41,42)(44,48)(45,47);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18)(19,23)(20,25)(21,22)(24,26)(27,30)(28,29)(31,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,21,26)(18,23,22,24)(20,27,28,32)(25,30,29,31)(33,35)(34,40)(36,47)(37,45)(38,46)(39,43)(41,48)(42,44);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19)(18,24)(20,27)(21,26)(22,23)(25,31)(28,32)(29,30)(33,36,38,42)(34,41,43,37)(35,44,46,47)(39,48,40,45);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,24)(20,25)(21,22)(23,26)(27,31)(28,29)(30,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,48)(45,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20,21,28)(18,25,22,29)(19,27,26,32)(23,30,24,31)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,47,40,44)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034534);