Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,21,25)(19,27,30,23)(20,31,33,24)(22,26,35,43)(28,38,46,45)(29,39,36,40)(32,41,47,48)(34,44,37,42);(1,2,5,6)(3,7,4,8)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,36,35,29)(24,41,31,48)(26,39,43,40)(28,37,46,34)(38,44,45,42);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20,21,33)(18,24,25,31)(19,28,30,46)(22,37,35,34)(23,38,27,45)(26,44,43,42)(29,47,36,32)(39,41,40,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(18,25)(20,33)(23,27)(26,43)(28,46)(32,47)(34,37)(39,40);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(18,25)(19,29)(20,33)(22,35)(23,39)(27,40)(28,32)(30,36)(38,48)(41,45)(42,44)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=gi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6647858);