Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,4)(5,6)(7,8)(17,18)(19,29)(20,26)(21,35)(22,27)(23,38)(24,31)(25,34)(28,37)(30,40)(32,41)(33,44)(36,43)(39,48)(42,46)(45,47);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,32)(25,42)(26,40)(27,29)(28,41)(34,48)(35,44)(36,46)(37,47)(38,45)(39,43);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,21)(18,25,27,43)(19,30,31,33)(23,36,37,34)(24,39,29,42)(26,28,35,38)(32,48,47,46)(40,45,44,41);(2,6)(3,8)(11,16)(13,15)(17,21,22,20)(18,26,27,35)(19,30,31,33)(23,36,37,34)(24,40,29,44)(25,28,43,38)(32,46,47,48)(39,45,42,41);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(20,21)(25,43)(26,35)(30,33)(34,36)(39,42)(40,44)(46,48)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6796460);