Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347226 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | D4×2−1+4 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,6,4,7)(11,17)(12,19)(14,22)(15,20)(18,21)(23,24);(2,6)(3,4)(7,8)(9,10,13,16)(11,14,18,23)(12,20,19,15)(17,24,21,22);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,19)(20,22);(2,7)(3,4)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72271566, #72272117 |
| maximal normal subgroup of | #72271566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254458);