Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,25)(17,19)(18,24)(20,27)(22,26)(23,28);(5,7)(6,9)(8,11)(10,12)(13,15,17,23)(14,18,19,22)(16,26,27,24)(20,28,25,21);(1,2,3,4)(7,11)(9,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,26)(24,28)(25,27);(7,11)(9,12)(14,19)(15,23)(20,25)(24,26);(6,10)(9,12)(14,19)(18,22)(20,25)(21,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380297);