Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | |
2235 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,22,37)(20,30,33,38)(21,34,35,26)(25,31,41,28)(29,43,36,39)(32,46,47,44)(40,42,48,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,35,36)(25,30,41,38)(26,39,34,43)(32,45,47,42)(40,44,48,46);(11,15)(13,16)(17,20,23,33)(18,25,27,41)(19,28,22,31)(21,32,35,47)(24,38,37,30)(26,40,34,48)(29,42,36,45)(39,46,43,44);(3,8)(5,7)(10,14)(11,15)(17,21,23,35)(18,26,27,34)(19,29,22,36)(20,32,33,47)(24,39,37,43)(25,40,41,48)(28,42,31,45)(30,44,38,46);(2,6)(5,7)(11,15)(13,16)(21,35)(26,34)(29,36)(32,47)(39,43)(40,48)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107464);