Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185463 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3517 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6,7,8)(9,10)(11,16,12,15)(13,17)(14,18)(19,23,20,22)(21,24);(1,2)(3,6)(4,8)(5,7)(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22);(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22);(1,3,5,4)(2,6,7,8)(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92670);