Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42683 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 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)(5,6,8,9)(7,11,12,10)(13,14,19,24)(15,20,28,36)(16,21,18,23)(17,22,31,39)(25,33,26,34)(27,35,42,44)(29,37,32,40)(30,38,41,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,30)(18,25)(19,28)(21,34)(22,38)(23,33)(24,36)(27,32)(29,42)(31,41)(35,40)(37,44)(39,43);(1,2,3,4)(6,9)(10,11)(15,25)(16,18)(20,33)(21,23)(26,28)(27,41)(29,32)(30,42)(34,36)(35,43)(37,40)(38,44);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,24,23)(15,26,28,25)(17,29,31,32)(20,34,36,33)(22,37,39,40)(27,30,42,41)(35,38,44,43);(13,17,19,31)(14,22,24,39)(15,27,28,42)(16,29,18,32)(20,35,36,44)(21,37,23,40)(25,41,26,30)(33,43,34,38)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591957);