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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,22)(17,23)(18,21)(19,24)(20,28)(26,34)(27,41)(29,36)(30,44)(31,39)(32,37)(33,40)(35,43)(38,42);(6,9)(10,12)(13,15)(14,20)(16,30)(17,29)(18,28)(19,26)(21,25)(22,38)(23,34)(24,36)(27,33)(31,43)(32,42)(35,40)(37,44)(39,41);(1,2,3,4)(14,21)(15,26)(17,19)(20,34)(22,37)(25,36)(27,42)(28,29)(30,43)(31,33)(35,38)(41,44);(13,16,18,32)(14,22,21,37)(15,27,28,43)(17,31,19,33)(20,35,25,41)(23,39,24,40)(26,42,29,30)(34,38,36,44);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,21,24)(15,26,28,29)(16,31,32,33)(20,36,25,34)(22,39,37,40)(27,42,43,30)(35,44,41,38)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,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, 6352521);