Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #42788 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2049 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)(7,10)(8,12)(9,11)(13,14)(15,26)(16,22)(17,34)(18,24)(19,25)(20,23)(21,30)(27,43)(28,38)(29,41)(31,44)(32,39)(33,42)(35,37)(36,40);(5,7,10,6)(8,12,9,11)(13,15)(14,21)(16,31)(17,28)(18,33)(19,30)(20,36)(22,24)(23,38)(25,26)(27,32)(29,35)(34,40)(37,39)(41,43)(42,44);(2,4)(13,16,20,32,19,33,17,29)(14,22,23,39,25,42,34,41)(15,27,28,31,30,35,36,18)(21,37,40,24,26,43,38,44);(5,8,10,9)(6,11,7,12)(13,17,19,20)(14,23,25,34)(15,28,30,36)(16,32,33,29)(18,35,31,27)(21,38,26,40)(22,41,42,39)(24,43,44,37);(5,9,10,8)(6,11,7,12)(13,18,20,27,19,31,17,35)(14,24,23,43,25,44,34,37)(15,29,28,16,30,32,36,33)(21,39,40,42,26,41,38,22)Polycyclic presentation:
2^9|b^2=fi,c^2=h,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718607);