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 |
| prime radical | #36066 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,21)(22,24);(1,2,6,4)(3,8,5,7)(9,11,14,12)(10,15,16,13)(17,19)(18,22)(20,23)(21,24);(1,3,6,5)(2,7,4,8)(9,12,14,11)(10,15,16,13)(17,20,21,18)(19,23,24,22);(1,2,6,4)(3,8,5,7)(9,13,14,15)(10,12,16,11)(17,19)(18,23)(20,22)(21,24);(1,4,6,2)(3,8,5,7);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7596943);