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 | |
10856 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)(3,4)(6,8)(9,10)(11,16)(12,13)(14,18)(15,19)(17,23)(20,21)(22,24);(5,6,7,8)(11,19)(15,16)(20,22)(21,24);(2,4)(9,11,14,19)(10,15,18,16)(12,22,17,20)(13,21,23,24);(1,2)(3,4)(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,22)(23,24);(9,12,14,17)(10,13,18,23)(11,20,19,22)(15,24,16,21);(9,13)(10,17)(11,21)(12,18)(14,23)(15,20)(16,22)(19,24)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581321);