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 | #44932 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(2,5)(3,8)(6,7)(14,23)(17,24);(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(2,5)(3,8)(6,7)(12,22)(15,20)(17,24)(19,21)Polycyclic presentation:
2^9|a^2=h,c^2=i,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 | #72257338, #72257370, #72257412, … |
| maximal normal subgroup of | #72257338, #72257370, #72257412, … |
| maximal quotient of | #72257943, #72264024, #72278915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152932);