Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 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)(5,6)(7,10)(8,12)(9,11)(13,14)(15,24)(16,19)(17,29)(18,30)(20,23)(21,27)(22,32)(25,34)(26,35)(28,37)(31,36)(33,43)(38,41)(39,42)(40,44);(5,7,10,6)(8,12,9,11)(13,15)(14,21)(16,24)(17,26)(18,28)(19,27)(20,34)(22,35)(23,37)(25,30)(29,38)(31,42)(32,41)(33,44)(36,40)(39,43);(2,4)(13,16,19,14)(15,25,27,37)(17,22,32,29)(18,30,20,23)(21,34,24,28)(26,40,41,39)(31,36,33,43)(35,42,38,44);(5,8,10,9)(6,11,7,12)(13,17,19,32)(14,22,16,29)(15,26,27,41)(18,31,20,33)(21,35,24,38)(23,36,30,43)(25,39,37,40)(28,42,34,44);(5,9,10,8)(6,11,7,12)(13,18)(14,23)(16,30)(17,31)(19,20)(21,24)(22,36)(28,34)(29,43)(32,33)(35,38)(42,44)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6583909);