Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,8,12,11)(13,14,17,19)(15,16,20,18)(22,25)(23,27);(2,4)(6,10)(8,11)(13,15,17,20)(14,18,19,16)(22,25)(26,28);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16,17,18)(14,15,19,20)(21,22,24,25)(23,26,27,28);(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20);(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281097, #72281440, #72281913 |
| maximal normal subgroup of | #72281097, #72281440, #72281913 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1021562);