Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(17,18,23,28,22,26,21,27)(19,29,33,39,31,24,32,40)(20,34,35,41,36,42,37,25)(30,44,46,38,45,48,47,43);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31)(18,24,26,29)(20,30,36,45)(21,32,23,33)(25,38,41,43)(27,39,28,40)(34,48,42,44)(35,46,37,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(39,48)(40,44);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(18,26)(19,31)(25,41)(27,28)(30,45)(32,33)(34,42)(46,47);(11,15)(14,16)(17,21,22,23)(18,27,26,28)(19,32,31,33)(20,35,36,37)(24,39,29,40)(25,34,41,42)(30,46,45,47)(38,48,43,44)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6457156);