Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,13)(4,11,12,8)(10,16,15,14)(17,18,22,25)(19,27,29,23)(20,26,21,24)(28,31,30,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,6)(7,13)(8,11)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(2,6)(3,9)(8,11)(10,15)(18,25)(19,29)(24,26)(28,30)(34,40)(38,41)(39,48)(46,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,20,22,21)(18,26,25,24)(19,28,29,30)(23,32,27,31);(17,21,22,20)(18,24,25,26)(19,30,29,28)(23,31,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000240);