Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #30413 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1508 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)(19,28)(20,32)(21,26)(22,23)(24,31)(25,33)(27,36)(29,39)(30,45)(34,41)(35,40)(37,44)(38,46)(42,43)(47,48);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(18,23)(19,29)(20,33)(26,27)(31,44)(34,35)(38,48)(42,45);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,31,22,29,36,44)(18,24,27,28,23,37,26,39)(20,30,35,47,33,43,34,46)(25,38,40,42,32,48,41,45);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,33)(23,32)(24,38)(26,41)(27,40)(28,42)(29,43)(31,47)(34,36)(37,48)(39,45)(44,46);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,22,36)(18,26,23,27)(19,31,29,44)(20,34,33,35)(24,39,37,28)(25,40,32,41)(30,46,43,47)(38,42,48,45)Polycyclic presentation:
2^9|b^2=fi,c^2=fgi,d^2=fi,e^2=f,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=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822995);