Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1501 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)(11,15)(14,16)(17,18)(19,24)(20,30)(21,34)(22,36)(23,27)(25,32)(26,37)(28,42)(29,44)(31,45)(33,46)(35,39)(38,43)(40,47)(41,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,31)(21,23)(22,29)(25,40)(26,39)(27,34)(28,33)(30,45)(32,47)(35,37)(36,44)(38,41)(42,46)(43,48);(10,13)(14,16)(17,20,22,32,23,33,37,48)(18,25,26,30,27,41,36,46)(19,28,29,43,21,31,35,47)(24,38,39,42,34,40,44,45);(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,24,27,34)(20,31,33,28)(22,35,37,29)(25,38,41,40)(26,39,36,44)(30,42,46,45)(32,47,48,43);(1,3,4,7)(2,6,5,8)(19,21)(24,34)(28,31)(29,35)(38,40)(39,44)(42,45)(43,47)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343258);