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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 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)(5,6,9,8)(7,10,12,11)(13,14,18,21)(15,23,27,39)(16,28,30,20)(17,32,25,22)(19,33,35,26)(24,40,29,38)(31,37,41,43)(34,42,36,44);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,17,26,18,27,25,33)(14,19,22,23,21,35,32,39)(16,29,31,44,30,24,41,42)(20,36,37,40,28,34,43,38);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16,18,30)(14,20,21,28)(15,24,27,29)(17,31,25,41)(19,34,35,36)(22,37,32,43)(23,38,39,40)(26,42,33,44);(1,3,4,2)(7,12)(10,11)(14,21)(16,30)(19,35)(22,32)(23,39)(24,29)(31,41)(42,44);(13,15,25,33,18,27,17,26)(14,19,32,39,21,35,22,23)(16,24,41,44,30,29,31,42)(20,34,43,40,28,36,37,38)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=hi,d^2=gi,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, 7511547);