Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2305 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)(7,10)(8,9)(11,12)(14,18)(16,20)(19,23)(21,26)(24,25)(27,28);(6,9)(10,12)(13,14,17,18)(15,19,22,23)(16,20,24,25)(21,26,27,28);(1,2,3,4)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(13,16)(14,20)(15,21)(17,24)(18,25)(19,26)(22,27)(23,28);(5,7,8,11)(6,10,9,12)(15,22)(19,23)(21,27)(26,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^c=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537439);