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 | |
6856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,20,22)(19,21,24,23);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(1,2)(3,6)(4,5)(7,8);(1,3)(2,6)(4,7)(5,8)Polycyclic presentation:
2^9|b^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72241104, #72241112, #72241136, … |
| maximal normal subgroup of | #72241104, #72241112, #72241136, … |
| maximal quotient of | #72335267, #72335306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000818);