Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,26,17,20,16,21)(15,22,25,19,23,28,24,27);(2,3)(13,15)(14,19)(16,25)(17,23)(18,24)(20,27)(21,28)(22,26);(1,2,4,3)(6,9)(10,12)(15,23)(19,27)(22,28)(24,25);(1,3,4,2)(14,20)(15,23)(21,26)(24,25);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,26)(22,28)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374567);