Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1586 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)(17,18,22,27,23,26,21,35)(19,24,31,38,29,39,30,45)(20,32,33,40,34,41,36,25)(28,42,44,48,43,47,46,37);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,42)(33,44)(34,43)(35,45)(36,46)(40,48)(41,47);(10,13)(14,16)(17,20,23,34)(18,25,26,40)(19,28,29,43)(21,33,22,36)(24,37,39,48)(27,41,35,32)(30,44,31,46)(38,47,45,42);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(18,26)(19,29)(25,40)(27,35)(28,43)(30,31)(32,41)(44,46);(11,15)(14,16)(17,21,23,22)(18,27,26,35)(19,30,29,31)(20,33,34,36)(24,38,39,45)(25,41,40,32)(28,44,43,46)(37,47,48,42)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452990);