Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52096 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,20,26,19,25,17,23)(15,27,28,22,24,21,16,18);(6,9)(7,11)(13,15)(14,21)(16,17)(18,26)(19,24)(20,28)(22,23)(25,27);(1,2,3,4)(13,16,19,28)(14,22,25,18)(15,17,24,20)(21,23,27,26);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,26)(15,28,24,16)(18,27,22,21);(6,9)(10,12)(13,18,20,27,19,22,17,21)(14,24,26,16,25,15,23,28)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311405);