Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,26,36)(16,21,29,40)(17,30,18,22)(24,33,42,43)(25,35,31,34)(27,37,41,44)(28,39,32,38);(1,2,4,3)(6,9)(10,12)(13,15)(14,20)(16,27)(17,25)(18,31)(19,26)(21,37)(22,34)(23,36)(24,28)(29,41)(30,35)(32,42)(33,39)(38,43)(40,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,38)(23,40)(25,41)(26,42)(27,31)(30,39)(34,37)(35,44)(36,43);(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,38)(23,40)(25,41)(26,42)(27,31)(30,39)(34,37)(35,44)(36,43);(13,17,19,18)(14,22,23,30)(15,25,26,31)(16,28,29,32)(20,34,36,35)(21,38,40,39)(24,41,42,27)(33,37,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347853);