Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,10,12,8)(13,14)(15,17)(16,19)(18,20)(21,22,26,24)(23,27,25,28);(5,7,9,12)(6,10,11,8)(13,15)(14,16)(17,20)(18,19)(21,23,26,25)(22,27,24,28);(6,11)(8,10)(13,16,18,17)(14,19,20,15)(21,24,26,22)(23,27,25,28);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,17,18,16)(14,15,20,19)(21,25,26,23)(22,27,24,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4052389);