Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,21,27,22,26,37,25)(19,28,32,42,31,23,30,39)(20,33,34,41,35,40,36,24)(29,44,47,38,46,48,45,43);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,23,26,28)(20,29,35,46)(21,32,37,30)(24,38,41,43)(25,42,27,39)(33,48,40,44)(34,47,36,45);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,35)(18,24,26,41)(19,29,31,46)(21,36,37,34)(23,38,28,43)(25,33,27,40)(30,47,32,45)(39,44,42,48);(2,6)(5,7)(11,16)(13,15)(17,21,22,37)(18,25,26,27)(19,30,31,32)(20,34,35,36)(23,39,28,42)(24,40,41,33)(29,45,46,47)(38,48,43,44);(10,14)(13,15)(18,26)(19,31)(24,41)(25,27)(29,46)(30,32)(33,40)(45,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6801435);