Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15278 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(9,10,12,14)(11,15,16,13)(17,18,20,23)(19,22,24,21)(25,26)(27,28)(29,31)(30,32);(2,3)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23)(25,27,29,32)(26,30,31,28);(4,8)(6,7)(10,14)(13,15)(19,24)(21,22)(27,32)(28,30);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,28)(29,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=ghi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263191, #72263193, #72263222, … |
| maximal normal subgroup of | #72263191, #72263193, #72263222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2086514);