Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,21,23,28)(16,25,22,19)(18,26,27,24);(2,3)(13,15,22,26,17,23,16,24)(14,18,25,28,20,27,19,21);(1,2,4,3)(6,9)(10,12)(15,23)(18,27)(21,28)(24,26);(1,3,4,2);(5,7,8,11)(6,10,9,12)(15,23)(18,27)(21,28)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374951);