Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,12,14,16,21,20)(11,15,19,17,18,22,24,23);(1,3)(2,5,6,7)(4,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24);(9,12,13,16,14,20,21,10)(11,17,19,22,18,23,24,15);(11,18)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,f^2=i,b^a=be,c^a=ci,g^a=gh,d^b=di,e^b=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72310612, #72310619, #72310625, … |
| maximal normal subgroup of | #72310612, #72310619, #72310625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89280);