Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,28)(20,32)(21,26)(22,23)(24,31)(25,33)(27,36)(29,39)(30,45)(34,41)(35,40)(37,44)(38,46)(42,43)(47,48);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,29)(20,33)(26,27)(31,44)(34,35)(38,48)(42,45);(2,6)(3,7)(10,13)(14,16)(17,19,21,31,22,29,36,44)(18,24,27,28,23,37,26,39)(20,30,35,47,33,43,34,46)(25,38,40,42,32,48,41,45);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,30)(21,35)(22,33)(23,32)(24,38)(26,41)(27,40)(28,42)(29,43)(31,47)(34,36)(37,48)(39,45)(44,46);(3,7)(5,8)(17,21,22,36)(18,26,23,27)(19,31,29,44)(20,34,33,35)(24,39,37,28)(25,40,32,41)(30,46,43,47)(38,42,48,45)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670729);