Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,21)(22,24)(26,28)(27,31);(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,26,29,28)(27,30,31,32);(11,16)(13,15)(18,20)(22,23)(26,28)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,23,24,22)(25,28,29,26)(27,32,31,30)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72262656, #72262660, #72262680, … |
| maximal normal subgroup of | #72262656, #72262660, #72262680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 983765);