Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42443 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14)(15,23)(16,17)(18,21)(19,22)(20,24)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,20,21,23)(16,25,27,22)(17,19,28,26);(2,4)(13,16,26,23,18,27,19,20)(14,17,25,15,21,28,22,24);(5,7,8,11)(6,10,9,12)(14,21)(16,27)(20,23)(22,25);(6,9)(10,12)(13,17,26,15,18,28,19,24)(14,16,25,23,21,27,22,20)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715138);