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 | |
6144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,6)(4,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24);(1,2,5,6)(3,7,8,4)(9,10,14,12)(11,13,15,16)(17,19,22,24)(18,23,20,21);(2,6)(3,8)(10,12)(11,15)(18,20)(19,24);(9,11)(10,13)(12,16)(14,15)(17,19,22,24)(18,23,20,21);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,16,15,13)(17,20,22,18)(19,21,24,23);(1,4,5,7)(2,3,6,8)(9,13,14,16)(10,15,12,11)(17,21)(18,19)(20,24)(22,23)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7617694);