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 | |
5553 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,4,6,7)(9,10)(11,15)(12,14)(13,16)(18,22)(23,24);(1,3)(2,4)(5,6)(7,8)(17,18)(19,23)(20,24)(21,22);(1,4)(2,6)(3,8)(5,7)(9,10,12,14)(11,13,16,15)(17,19,21,20)(18,23,22,24);(1,3)(2,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16);(1,3)(2,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,19)(18,24,22,23);(1,4)(2,6)(3,8)(5,7)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875144);