Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2061 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)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,27,37)(16,20,28,34)(17,30,25,21)(19,31,33,26)(24,36,38,44)(29,41,39,35)(32,42,43,40);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,26,18,27,25,31)(14,19,21,23,22,33,30,37)(16,24,29,40,28,38,39,42)(20,32,35,36,34,43,41,44);(1,2,4,3)(6,9)(10,11)(13,16,18,28)(14,20,22,34)(15,24,27,38)(17,29,25,39)(19,32,33,43)(21,35,30,41)(23,36,37,44)(26,40,31,42);(1,3,4,2)(5,7,8,12)(6,10,9,11)(16,28)(20,34)(24,38)(29,39)(32,43)(35,41)(36,44)(40,42);(13,15,25,31,18,27,17,26)(14,19,30,37,22,33,21,23)(16,24,39,42,28,38,29,40)(20,32,41,44,34,43,35,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504108);