Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #385 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,13,16,14,17,12,22)(11,18,21,23,20,24,19,15);(2,4)(9,11)(10,15,17,23)(12,19)(13,21)(14,20)(16,18,22,24);(5,6,7,8)(11,19,20,21)(15,18,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235150, #72238816, #72238844 |
| maximal normal subgroup of | #72235150, #72238816, #72238844 |
| maximal quotient of | #72235177 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31143);