Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 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)(13,14)(15,26)(16,22)(17,23)(18,24)(19,34)(20,25)(21,28)(27,41)(29,36)(30,44)(31,39)(32,40)(33,37)(35,43)(38,42);(5,6)(7,11)(8,10)(9,12)(13,15,18,33,20,29,19,28)(14,21,34,36,25,37,24,26)(16,30,32,43,17,27,31,42)(22,38,39,41,23,35,40,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,17)(14,22,25,23)(15,27,29,30)(18,32,19,31)(21,35,37,38)(24,40,34,39)(26,41,36,44)(28,43,33,42);(7,12)(9,11)(13,17,20,16)(14,23,25,22)(15,27,29,30)(18,31,19,32)(21,35,37,38)(24,39,34,40)(26,41,36,44)(28,43,33,42);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,28,29,33)(16,31,17,32)(21,36,37,26)(22,39,23,40)(27,42,30,43)(35,41,38,44)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410206);