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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,18,23)(15,24,28,20)(16,21,31,39)(17,32,19,22)(25,36,27,34)(26,41,43,35)(29,44,42,37)(30,40,33,38);(6,9)(10,12)(13,15,18,28)(14,20,23,24)(16,29,31,42)(17,27,19,25)(21,37,39,44)(22,34,32,36)(26,33,43,30)(35,38,41,40);(2,4)(5,7,8,11)(6,10,9,12)(15,25)(17,19)(20,34)(22,32)(24,36)(26,29)(27,28)(30,33)(35,37)(38,40)(41,44)(42,43);(13,16)(14,21)(15,26)(17,30)(18,31)(19,33)(20,35)(22,38)(23,39)(24,41)(25,29)(27,42)(28,43)(32,40)(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=f,b^2=i,c^2=fi,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, 6345671);