Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(11,15)(14,16)(17,18)(19,23)(20,21)(22,24)(26,28)(27,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,26,29,28)(27,30,32,31);(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|b^2=fghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232639, #72232640, #72232674, … |
| maximal normal subgroup of | #72232639, #72232640, #72232674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2629403);