Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14)(15,23)(16,28)(17,21)(18,22)(19,26)(20,29)(24,34)(25,33)(27,32)(30,41)(31,36)(35,42)(37,40)(38,43)(39,44);(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,25)(17,26)(18,24)(20,33)(21,23)(22,32)(27,31)(28,37)(29,40)(30,38)(34,36)(35,43)(39,42)(41,44);(1,2,3,4)(15,24)(16,29)(18,31)(19,32)(20,28)(22,36)(23,34)(25,39)(26,27)(33,44)(37,43)(38,40);(5,7,9,8)(6,10,11,12)(13,16,17,29)(14,20,21,28)(15,25,26,40)(18,30,31,42)(19,33,23,37)(22,35,36,41)(24,38,27,39)(32,43,34,44);(5,8,9,7)(6,10,11,12)(14,21)(15,24)(18,31)(19,34)(20,28)(23,32)(25,38)(26,27)(30,42)(33,44)(37,43)(39,40)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6593989);