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 | |
6268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,13)(12,14)(15,16);(1,3,6,5)(2,7,8,4)(9,11)(10,13)(12,16)(14,15)(18,21)(22,24);(2,8)(4,7)(10,12)(11,15)(17,18,20,21)(19,22,23,24);(2,8)(3,5)(11,15)(13,16)(19,23)(22,24);(1,4,6,7)(2,3,8,5)(9,12,14,10)(11,13,15,16)(17,18)(19,22)(20,21)(23,24);(1,5,6,3)(2,7,8,4)(9,13)(10,11)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72249400, #72249491, #72249493, … |
| maximal normal subgroup of | #72249400, #72249491, #72249493, … |
| maximal quotient of | #72263278, #72263390, #72263467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506875);