Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1604 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)(3,7)(4,6)(5,8)(17,18,23,35,22,27,19,24)(20,30,28,25,33,37,32,40)(21,26,36,46,34,41,29,38)(31,43,42,39,44,47,45,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,35)(20,28,33,32)(21,29,34,36)(25,37,40,30)(26,38,41,46)(31,42,44,45)(39,47,48,43);(9,11)(10,13)(12,16)(14,15)(17,20,22,33)(18,25,27,40)(19,28,23,32)(21,31,34,44)(24,37,35,30)(26,39,41,48)(29,42,36,45)(38,47,46,43);(3,8)(5,7)(10,14)(11,16)(17,21,22,34)(18,26,27,41)(19,29,23,36)(20,31,33,44)(24,38,35,46)(25,39,40,48)(28,42,32,45)(30,43,37,47);(2,6)(5,7)(10,14)(13,15)(21,34)(26,41)(29,36)(31,44)(38,46)(39,48)(42,45)(43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7216430);