Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,7,8)(14,19)(15,22)(17,21)(24,28);(2,3)(10,12)(13,14,18,19)(15,23,22,20)(16,17,26,21)(24,27,28,25);(9,10,11,12)(13,15,18,22)(14,20,19,23)(16,24,26,28)(17,25,21,27);(2,3)(5,6)(7,8)(13,16,18,26)(14,21,19,17)(15,24,22,28)(20,27,23,25);(1,3,4,2)(13,17,18,21)(14,16,19,26)(15,25,22,27)(20,24,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2723336);