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 | |
11386 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)(11,15)(12,21)(13,17)(14,18)(16,22)(19,23)(20,24);(5,6,7,8)(9,11)(10,15)(12,13)(14,20)(16,23)(17,21)(18,24)(19,22);(2,4)(9,12,14,22)(10,16,18,21)(11,19,20,13)(15,17,24,23);(1,2)(3,4)(9,12)(10,16)(11,13)(14,22)(15,23)(17,24)(18,21)(19,20);(9,13)(10,17)(11,12)(14,19)(15,21)(16,24)(18,23)(20,22);(9,10)(11,15)(12,16)(13,23)(14,18)(17,19)(20,24)(21,22)Polycyclic presentation:
2^9|b^2=g,c^2=i,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, 7603631);