Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14)(15,25)(16,22)(17,23)(18,21)(19,24)(20,28)(26,34)(27,41)(29,36)(30,44)(31,39)(32,37)(33,40)(35,42)(38,43);(5,7,9,8)(6,10,12,11)(13,15,18,28)(14,20,21,25)(16,30,32,43)(17,29,19,26)(22,38,37,44)(23,34,24,36)(27,33,42,31)(35,40,41,39);(1,2,3,4)(5,7,9,8)(6,11,12,10)(14,21)(15,26)(17,19)(20,34)(22,37)(25,36)(27,30)(28,29)(31,33)(35,44)(38,41)(42,43);(5,8,9,7)(6,10,12,11)(13,16,18,32)(14,22,21,37)(15,27,28,42)(17,31,19,33)(20,35,25,41)(23,39,24,40)(26,30,29,43)(34,44,36,38);(13,17,18,19)(14,23,21,24)(15,26,28,29)(16,31,32,33)(20,36,25,34)(22,39,37,40)(27,30,42,43)(35,38,41,44)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351839);