Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(11,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(1,3,4,7)(2,5,6,8)(10,13)(14,16)(18,19)(21,29)(23,25)(27,30)(35,42)(36,45)(38,41)(39,44);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,32,30,31)(33,35)(34,38)(36,43)(37,42)(39,46)(40,41)(44,47)(45,48);(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=gi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5690957);