Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | #14757 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,24,23,21)(26,27)(28,32);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,16,15)(17,19)(18,21)(20,24)(22,23)(26,27)(30,31);(3,8)(5,7)(10,13)(14,15)(19,23)(21,24)(25,26,29,27)(28,30,32,31);(1,2)(3,7)(4,6)(5,8)(9,11,12,16)(10,14,13,15)(17,20,22,18)(19,21,23,24)(25,27,29,26)(28,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,21)(18,23)(19,20)(22,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=fh,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3880901);