Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| 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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14)(15,22)(16,21)(17,19)(18,25)(20,24)(23,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(21,25)(24,26);(2,4)(13,15)(14,20)(16,24)(17,23)(18,26)(19,27)(21,28)(22,25);(5,7,8,11)(6,10,9,12)(14,19)(20,27)(21,25)(22,28);(2,4)(6,9)(10,12)(13,16,17,18)(14,21,19,25)(15,24,23,26)(20,28,27,22)Polycyclic presentation:
2^9|d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577457);