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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16810 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,10)(7,12)(13,14,16,18)(15,19,20,17)(22,26)(23,24);(2,4)(5,6,9,10)(7,11,12,8)(15,20)(17,19)(21,22,25,26)(23,27,24,28);(1,2)(3,4)(13,15)(14,17)(16,20)(18,19)(21,23,25,24)(22,27,26,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,27)(24,28)(25,26);(5,8,9,11)(6,12,10,7)(13,14)(15,17)(16,18)(19,20)(21,24,25,23)(22,27,26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgi,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, 2795456);