Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18,22,28)(19,29,32,24)(20,26,35,41)(21,27,23,37)(25,36,42,34)(30,44,47,39)(31,45,33,40)(38,46,43,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32)(18,24,28,29)(20,30,35,47)(21,31,23,33)(25,38,42,43)(26,39,41,44)(27,40,37,45)(34,46,36,48);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,35)(18,25,28,42)(19,30,32,47)(21,34,23,36)(24,38,29,43)(26,27,41,37)(31,46,33,48)(39,40,44,45);(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,39)(25,37)(27,42)(28,41)(29,44)(31,48)(32,47)(33,46)(38,45)(40,43);(10,13)(14,16)(17,21,22,23)(18,27,28,37)(19,31,32,33)(20,34,35,36)(24,40,29,45)(25,41,42,26)(30,46,47,48)(38,44,43,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421289);