Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
10730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,16,20)(11,12,13,17)(14,18,22,23)(15,19,24,21);(5,6)(7,8)(9,11,16,13)(10,17,20,12)(14,21,22,19)(15,18,24,23);(2,3)(9,12,16,17)(10,11,20,13)(14,15,22,24)(18,21,23,19);(1,3,4,2)(9,13,16,11)(10,12,20,17)(14,19,22,21)(15,23,24,18);(9,14)(10,18)(11,21)(12,15)(13,19)(16,22)(17,24)(20,23);(9,15)(10,19)(11,18)(12,22)(13,23)(14,17)(16,24)(20,21)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606269);