Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1539 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)(17,18,23,27)(19,28,31,39)(20,25,32,41)(21,26,22,37)(24,35,40,30)(29,45,33,48)(34,43,36,44)(38,46,42,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,32)(25,41)(29,33)(34,36)(38,42)(43,44)(45,48)(46,47);(2,6)(5,7)(10,13)(14,16)(17,19,21,35,23,31,22,30)(18,24,37,39,27,40,26,28)(20,33,36,47,32,29,34,46)(25,42,43,45,41,38,44,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,23,32)(18,25,27,41)(19,29,31,33)(21,36,22,34)(24,38,40,42)(26,44,37,43)(28,45,39,48)(30,47,35,46);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,37)(19,30,31,35)(20,34,32,36)(24,39,40,28)(25,43,41,44)(29,46,33,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=hi,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,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6692087);