Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,8)(13,14)(15,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28);(10,12)(14,15)(19,21)(20,22)(27,28);(1,2,3,4)(5,6,7,8)(9,10,11,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)Polycyclic presentation:
2^9|c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424789);