Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2978 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)(5,6)(7,8)(9,10)(11,15,19,23)(12,16)(13,18)(14,17)(20,24,21,22);(2,4)(9,11,13,20,14,19,12,21)(10,15,16,22,17,23,18,24);(6,8)(10,16)(11,19)(12,13)(15,22)(17,18)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72236779, #72237077, #72237504, … |
| maximal normal subgroup of | #72236779, #72237077, #72237504, … |
| maximal quotient of | #72226736, #72236795, #72237081 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30840);