Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#2407 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3824 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)(3,4)(6,8)(13,14)(15,18)(16,21)(17,22)(19,23)(20,25)(24,27)(26,28);(1,2,3,4)(10,12)(14,15)(16,23)(17,25)(27,28);(9,10)(11,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,28,26,27);(1,2,3,4)(5,6)(7,8)(13,16)(14,19)(15,21)(17,26)(18,23)(20,28)(22,27)(24,25);(1,2)(3,4)(13,17)(14,20)(15,22)(16,24)(18,25)(19,27)(21,28)(23,26)Polycyclic presentation:
2^9|b^2=fg,c^2=g,d^2=fg,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, 2713263);