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 | |
11114 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)(9,10,14,16)(11,13,18,15)(12,17,20,23)(19,21,24,22);(5,6,7,8)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,22,23,21);(2,3)(10,16)(11,18)(12,20)(21,22);(1,3,4,2)(9,11)(10,13)(12,19)(14,18)(15,16)(17,21)(20,24)(22,23);(9,12,14,20)(10,17,16,23)(11,19,18,24)(13,21,15,22);(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=hi,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, 7581562);