Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,23,21)(19,24,29,27)(20,30,28,25)(22,31,32,26);(2,5)(3,7)(10,11)(14,15)(18,21)(24,27)(25,30)(26,31);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,26,30,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,23,28)(18,25,21,30)(19,22,29,32)(24,26,27,31);(17,21,23,18)(19,27,29,24)(20,30,28,25)(22,31,32,26);(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,30)(25,27)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219287, #72267104, #72267106, … |
| maximal normal subgroup of | #72219285, #72219287, #72267104, … |
| maximal quotient of | #72219277 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000490);