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 | |
6912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,13)(14,15);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,21)(22,24);(2,6)(3,8)(9,10,12,13)(11,14,16,15)(18,21)(22,23);(17,18)(19,22)(20,21)(23,24);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,4,5,7)(2,3,6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|c^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265188, #72265477, #72265489, … |
| maximal normal subgroup of | #72265188, #72265477, #72265489, … |
| maximal quotient of | #72266525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7641207);