Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,22)(21,24);(1,2)(3,8)(4,7)(5,6)(17,19)(18,21)(20,24)(22,23);(2,6)(7,8)(10,13)(14,15)(17,18,20,22)(19,21,24,23);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23);(1,2)(3,7)(4,8)(5,6)Polycyclic presentation:
2^9|c^2=g,d^2=i,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72241099, #72241110, #72241135, … |
| maximal normal subgroup of | #72241099, #72241110, #72241135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7665169);