Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2283 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)(15,25)(16,20)(17,24)(18,28)(19,23)(21,26)(22,27);(5,7,8,12)(6,9,10,11)(13,15,19,16)(14,20,23,25)(17,27,26,18)(21,22,24,28);(2,4)(7,12)(9,11)(13,16,17,27,19,15,26,18)(14,20,24,22,23,25,21,28);(6,10)(9,11)(13,17,19,26)(14,21,23,24)(15,27,16,18)(20,22,25,28);(13,18,26,15,19,27,17,16)(14,22,21,20,23,28,24,25)Polycyclic presentation:
2^9|b^2=f,c^2=fhi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717713);