Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #393 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1850 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)(11,14,18,22)(12,16)(13,15)(17,19,23,21)(20,24);(2,4)(9,11)(10,14,15,22)(12,19)(13,18)(16,23,24,17)(20,21);(5,6,7,8)(9,10)(11,17,18,23)(12,16)(13,15)(14,21,22,19)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,f^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gi,d^c=di,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72211251, #72235150, #72235493, … |
| maximal normal subgroup of | #72211251, #72235150, #72235493, … |
| maximal quotient of | #72211196, #72235161, #72303631 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31116);