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 | #42788 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2158 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)(6,9)(7,11)(13,14)(15,26)(16,22)(17,34)(18,24)(19,25)(20,23)(21,30)(27,44)(28,37)(29,41)(31,43)(32,42)(33,40)(35,39)(36,38);(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,31)(17,28)(18,32)(19,30)(20,36)(22,39)(23,37)(24,40)(25,26)(27,33)(29,35)(34,38)(41,43)(42,44);(2,4)(13,16,19,29)(14,22,25,41)(15,27,30,18)(17,33,20,32)(21,24,26,44)(23,42,34,40)(28,35,36,31)(37,39,38,43);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,34)(15,28,30,36)(16,32,29,33)(18,35,27,31)(21,37,26,38)(22,40,41,42)(24,43,44,39);(6,9)(10,12)(13,18,19,27)(14,24,25,44)(15,29,30,16)(17,35,20,31)(21,22,26,41)(23,43,34,39)(28,33,36,32)(37,40,38,42)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,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, 6578701);