Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,23,18,15,17,21)(16,22,26,27,25,20,24,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,14)(16,20,25,22)(17,21,19,23)(24,28,26,27);(2,4)(6,9)(10,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28);(2,4)(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,15,23)(16,24,25,26)(20,27,22,28);(16,25)(20,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357106);