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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(34,35)(36,43)(37,45)(47,48);(2,5)(6,8)(11,15)(13,16)(18,22)(23,25)(24,30)(31,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,48,46,47);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,48,46,47);(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,28,29,26)(24,32,30,31)(33,36)(34,39)(35,41)(37,46)(38,43)(40,48)(42,47)(44,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,47)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=fgi,d^2=fg,e^2=fg,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, 5623168);