Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4150 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)(3,11)(4,9)(5,10)(6,8)(7,13)(12,16)(14,15)(17,18,22,23)(19,27,29,24)(20,26,21,25)(28,31,30,32)(33,34,37,39)(35,38,41,46)(36,40,44,47)(42,45,43,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,16)(10,15)(18,23)(20,21)(24,27)(28,30)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,27)(25,32)(26,31)(34,39)(35,41)(36,44)(45,48);(2,8)(7,11)(9,10)(15,16)(17,20,22,21)(18,25,23,26)(19,28,29,30)(24,31,27,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,15,11,16)(17,21,22,20)(18,26,23,25)(19,28,29,30)(24,31,27,32)(34,39)(35,41)(40,47)(42,43);(1,5,6,4)(2,10,8,9)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012811);