Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35328 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,26)(21,27)(22,36)(23,25)(24,31)(29,43)(30,37)(32,42)(33,40)(34,39)(35,41)(38,44)(45,48)(46,47);(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,32,21,30)(18,24,27,28,25,37,36,42)(20,29,33,44,34,45,35,46)(26,38,41,43,39,47,40,48);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(18,25)(24,37)(26,39)(27,36)(28,42)(38,47)(40,41)(43,48);(11,15)(13,16)(17,20,23,34)(18,26,25,39)(19,29,32,45)(21,35,22,33)(24,38,37,47)(27,41,36,40)(28,43,42,48)(30,46,31,44);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,21,23,22)(18,27,25,36)(19,30,32,31)(20,33,34,35)(24,28,37,42)(26,40,39,41)(29,44,45,46)(38,48,47,43)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384183);