Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2107 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);(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,16,20)(17,26,27,21)(19,25,28,24);(2,4)(13,15,18,16)(14,19,22,28)(17,25,27,24)(20,26,23,21);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,20,22,23)(17,25,27,24)(19,26,28,21);(6,9)(10,12)(14,21)(17,27)(19,20)(22,26)(23,28)(24,25)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577318);