Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(20,25)(21,26)(22,36)(23,27)(24,31)(29,39)(30,43)(32,40)(33,42)(34,37)(35,41)(38,46)(44,48)(45,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(19,29)(24,37)(28,39)(30,44)(31,34)(38,47)(43,48)(45,46);(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,29,21,34)(18,24,26,28,27,37,36,39)(20,30,32,46,33,44,35,45)(25,38,41,43,42,47,40,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,23,33)(18,25,27,42)(19,30,29,44)(21,35,22,32)(24,38,37,47)(26,41,36,40)(28,43,39,48)(31,46,34,45);(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,31,29,34)(20,32,33,35)(24,39,37,28)(25,40,42,41)(30,45,44,46)(38,43,47,48)Polycyclic presentation:
2^9|c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6414304);