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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2330 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)(13,14,19,23)(15,20,27,38)(16,21,30,42)(17,31,18,22)(24,34,25,35)(26,37,32,36)(28,39,43,44)(29,41,33,40);(6,9)(10,12)(13,15)(14,20)(16,28)(17,26)(18,32)(19,27)(21,39)(22,36)(23,38)(24,29)(25,33)(30,43)(31,37)(34,41)(35,40)(42,44);(2,4)(13,16)(14,21)(15,24)(17,29)(18,33)(19,30)(20,34)(22,40)(23,42)(25,27)(26,43)(28,32)(31,41)(35,38)(36,39)(37,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,29)(18,33)(19,30)(20,35)(22,40)(23,42)(24,27)(26,28)(31,41)(32,43)(34,38)(36,44)(37,39);(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,30,33)(20,36,38,37)(21,40,42,41)(24,43,25,28)(34,39,35,44)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=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346624);