Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,26)(17,20)(18,21)(22,23)(24,27)(25,28);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,24)(17,22)(18,25)(20,23)(21,28)(26,27);(1,2,4,3)(6,10)(9,12)(15,22)(16,17)(19,23)(20,26)(24,25)(27,28);(1,3,4,2)(13,14)(15,23)(16,20)(17,26)(18,21)(19,22)(24,28)(25,27);(13,16,18,17)(14,20,21,26)(15,22,25,24)(19,27,28,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348880);