Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,21)(22,24)(25,26)(27,28)(29,30)(31,32);(2,6)(3,7)(9,11,12,16)(10,13,14,15)(17,18,20,21)(19,23,24,22)(26,30)(27,31);(3,7)(4,8)(10,14)(13,15)(18,21)(22,23)(25,27,29,31)(26,28,30,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,16)(13,14)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,31)(30,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72334258 |
| maximal normal subgroup of | #72334258 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5156764);