Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,18,16,19,22,17,21)(15,23,26,25,27,24,28,20);(6,9)(10,12)(13,15)(14,20)(16,24)(17,26)(18,28)(19,27)(21,23)(22,25);(1,2,3,4)(13,14,17,21,19,22,18,16)(15,24,28,25,27,23,26,20);(5,7,8,11)(6,10,9,12)(13,16,18,22,19,21,17,14)(15,25,26,24,27,20,28,23);(13,17,19,18)(14,21,22,16)(15,26,27,28)(20,23,25,24)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367164);