Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(11,15)(14,16)(17,18,22,25)(19,23,28,39)(20,30,33,42)(21,34,36,26)(24,35,41,32)(27,43,31,48)(29,46,45,38)(37,47,40,44);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,25,39)(20,31,33,27)(21,29,36,45)(24,40,41,37)(26,38,34,46)(30,48,42,43)(32,47,35,44);(3,8)(5,7)(10,13)(14,16)(17,20,22,33)(18,24,25,41)(19,27,28,31)(21,35,36,32)(23,37,39,40)(26,42,34,30)(29,47,45,44)(38,48,46,43);(2,6)(5,7)(10,13)(14,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,37)(25,39)(26,38)(30,43)(31,33)(32,44)(34,46)(35,47)(36,45)(40,41)(42,48);(9,11,12,15)(10,14,13,16)(18,25)(19,28)(24,41)(26,34)(27,31)(29,45)(30,42)(44,47)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6560789);