Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | |
1579 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)(19,25)(20,27)(21,35)(22,28)(23,38)(24,29)(26,34)(30,40)(31,46)(32,41)(33,43)(36,45)(37,42)(39,48)(44,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,33)(18,25,29,43)(20,31,21,30)(22,37,23,32)(26,44,45,39)(27,46,35,40)(28,42,38,41)(34,47,36,48);(10,13)(14,16)(17,20,24,21)(18,26,29,45)(19,30,33,31)(22,34,23,36)(25,39,43,44)(27,28,35,38)(32,47,37,48)(40,42,46,41);(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,27,29,35)(19,31,33,30)(22,34,23,36)(25,40,43,46)(26,28,45,38)(32,47,37,48)(39,42,44,41);(1,3,4,7)(2,6,5,8)(17,22,24,23)(18,28,29,38)(19,32,33,37)(20,34,21,36)(25,41,43,42)(26,35,45,27)(30,47,31,48)(39,40,44,46)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342583);