Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,14,18,15,11,13,17)(12,20,21,16,23,24,22,19);(5,6,7,8)(9,11,15,10)(12,19,23,16)(13,18,14,17)(20,22,24,21);(1,2,3,4)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(9,13,15,14)(10,17,11,18)(12,21,23,22)(16,24,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,f^c=fi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239302, #72239396, #72258245, … |
| maximal normal subgroup of | #72239302, #72239396, #72258245, … |
| maximal quotient of | #72235046, #72239189, #72239190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261242);