Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 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)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,41)(30,37)(32,46)(36,48)(38,47)(39,43)(40,44)(42,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,45)(32,42)(33,43)(34,44)(35,40)(38,39)(41,46)(47,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,21,32,34,47)(18,24,26,31,25,38,35,46)(19,28,30,43,29,42,44,48)(23,36,37,45,27,39,40,41);(2,6)(5,7)(11,16)(13,15)(18,25)(19,29)(20,32)(26,35)(30,44)(33,47)(36,39)(41,45);(10,14)(13,15)(20,32)(24,38)(28,42)(31,46)(33,47)(36,39)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=fi,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815781);