Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,22,23)(12,16)(13,17)(14,18)(15,21,24,20);(2,6,8,7)(9,11)(10,15,18,24)(12,20)(13,21)(14,22)(16,23,17,19);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19);(1,3,5,4)(2,6,8,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=hi,f^2=h,b^a=be,e^a=eh,g^a=ghi,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72211165, #72211196, #72211214, … |
| maximal normal subgroup of | #72211165, #72211196, #72211214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88339);