Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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 | C2×#3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2475 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)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,26)(24,27)(25,28);(6,9)(10,12)(15,23)(16,25)(17,24)(19,26)(20,28)(22,27);(2,4)(13,15,18,25)(14,19,21,28)(16,24,23,17)(20,27,26,22);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,24)(17,25)(18,23)(19,27)(21,26)(22,28);(14,21)(19,28)(20,26)(22,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308156, #72308238, #72308276 |
| maximal normal subgroup of | #72308156, #72308238, #72308276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346621);