Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 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)(3,6)(4,5)(7,8)(17,18,23,28)(19,24,31,40)(20,33,21,26)(22,36,37,27)(25,35,42,34)(29,43,30,39)(32,47,46,41)(38,45,48,44);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,28,40)(20,29,21,30)(22,32,37,46)(25,38,42,48)(26,39,33,43)(27,41,36,47)(34,44,35,45);(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,23)(22,35)(24,38)(26,27)(28,42)(30,31)(32,45)(33,36)(34,37)(39,41)(40,48)(43,47)(44,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,21,23,20)(18,26,28,33)(19,30,31,29)(22,35,37,34)(24,39,40,43)(25,36,42,27)(32,45,46,44)(38,47,48,41);(2,5)(6,8)(10,14)(13,15)(19,31)(20,21)(24,40)(25,42)(26,33)(32,46)(34,35)(41,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7143466);