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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,24)(16,21)(17,32)(18,23)(19,22)(20,28)(25,36)(26,41)(27,34)(29,44)(30,40)(31,39)(33,38)(35,43)(37,42);(6,9)(10,12)(13,15)(14,20)(16,29)(17,27)(18,28)(19,25)(21,37)(22,34)(23,24)(26,30)(31,42)(32,36)(33,43)(35,40)(38,41)(39,44);(2,4)(15,25)(17,19)(20,34)(22,32)(24,36)(26,29)(27,28)(30,33)(35,37)(38,40)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16,18,31)(14,21,23,39)(15,26,28,43)(17,30,19,33)(20,35,24,41)(22,38,32,40)(25,29,27,42)(34,37,36,44);(13,17,18,19)(14,22,23,32)(15,27,28,25)(16,30,31,33)(20,34,24,36)(21,38,39,40)(26,42,43,29)(35,37,41,44)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6346660);