Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245711 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #45355 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,19,21,24)(12,17,13,16)(15,22,23,20);(2,5)(3,8)(6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,19)(18,23);(10,16)(12,13)(15,19)(17,18)(20,22)(23,24);(2,5)(3,8)(6,7)(10,17)(12,13)(15,19)(16,18)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213883, #72230977, C2×#245712, … |
| maximal normal subgroup of | #72213883, #72230977, C2×#245712, … |
| maximal quotient of | C2×#245712, #72240389, #72257933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152908);