Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1601 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)(3,5)(4,6)(7,8)(10,15,16,23)(11,17)(12,18,22,21)(19,20);(1,3,4,7)(2,5,6,8)(9,10,14,16)(11,18,17,21)(12,19,22,20)(13,23,24,15);(2,6)(5,8)(9,11)(10,12)(13,19)(14,17)(15,21)(16,22)(18,23)(20,24);(2,6)(5,8)(9,12)(10,11)(13,18)(14,22)(15,20)(16,17)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72228066, #72228068 |
| maximal normal subgroup of | #72228066, #72228068 |
| maximal quotient of | #72209991, #72238301, #72238811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291054);