Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | #37273 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(18,22)(23,25)(24,28)(30,31);(10,14)(11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(10,14)(13,15)(18,22)(19,26)(24,28)(27,32);(1,3,5,4)(2,7,6,8)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542065);