Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,10,8,12)(13,14,18,23,20,25,19,24)(15,26,31,21,28,40,16,29)(17,22,32,37,34,39,33,38)(27,36,43,35,41,44,30,42);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,18)(17,30)(19,31)(20,28)(22,36)(23,26)(24,40)(25,29)(27,33)(32,41)(34,43)(35,38)(37,42)(39,44);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,15,20,31,19,28)(14,21,23,40,25,29,24,26)(17,30,32,27,34,43,33,41)(22,35,37,44,39,42,38,36);(2,4)(13,17,19,33,20,34,18,32)(14,22,24,38,25,39,23,37)(15,27,16,30,28,41,31,43)(21,35,26,36,29,42,40,44);(13,18,20,19)(14,23,25,24)(15,16,28,31)(17,32,34,33)(21,26,29,40)(22,37,39,38)(27,30,41,43)(35,36,42,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355318);