Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2439 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,19)(15,20,24,26)(16,21,25,27)(18,22,23,28);(5,7)(6,9)(8,12)(10,11)(14,19)(15,16)(18,23)(20,27)(21,26)(24,25);(2,4)(7,12)(9,11)(13,15,23,25,17,24,18,16)(14,20,28,27,19,26,22,21);(6,10)(9,11)(14,19)(20,26)(21,27)(22,28);(13,16,18,24,17,25,23,15)(14,21,22,26,19,27,28,20)Polycyclic presentation:
2^9|a^2=i,c^2=fh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717063);