Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| 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 | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3252 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,12,16,14)(17,18,21,22)(19,23,24,20)(26,27)(28,32);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(26,27)(30,31);(3,7)(6,8)(10,15)(12,14)(18,22)(20,23)(25,26,29,27)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,23,22,20)(25,27,29,26)(28,31,32,30);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,11,15,16)(17,20,21,23)(18,24,22,19)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=hi,c^2=ghi,d^2=h,e^2=fghi,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 | #72232597, #72232605, #72232622, … |
| maximal normal subgroup of | #72232597, #72232605, #72232622, … |
| maximal quotient of | #72328434, #72328676, #72328820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456233);