Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| 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 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2360 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)(7,12)(8,10)(9,11)(13,14,18,23,19,22,17,21)(15,24,26,16,27,28,25,20);(7,8)(10,12)(13,15,19,27)(14,20,22,16)(17,26,18,25)(21,24,23,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16,19,20)(14,15,22,27)(17,24,18,28)(21,25,23,26);(5,8,9,7)(6,10,11,12)(13,17,19,18)(14,21,22,23)(15,25,27,26)(16,24,20,28);(14,22)(16,20)(21,23)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6370206);