Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,22)(23,24)(26,30)(27,31);(2,3)(4,8)(10,13)(14,15)(17,18,20,22)(19,21,23,24)(25,26)(27,28)(29,30)(31,32);(2,3)(6,7)(11,16)(14,15)(19,23)(21,24)(26,30)(27,31);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,20,23)(18,21,22,24)(25,27,29,31)(26,28,30,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72300713, #72300724, #72300790 |
| maximal normal subgroup of | #72300713, #72300724, #72300790 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1490558);