Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93191 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C22×D4 | C2×C8 |
| Frattini, cosocle | #1438 | C22 |
| prime radical | #3484 | C4 |
| LCS | G ▷ C22×D4 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ #1438 ◁ G | |
| LECS | G ▷ #1438 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1 | |
391 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,13,14,23,15,28,30,7)(4,18,21,8)(5,12)(6,11)(9,20,26,32)(10,22)(16,29,31,25,17,27,19,24);(1,3)(2,7)(4,19)(5,17)(6,16)(8,13)(9,27)(10,23)(11,24)(12,25)(14,20)(15,22)(18,29)(21,30)(26,28)(31,32)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=dei,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72267805, #72267849 |
| maximal normal subgroup of | #72267805, #72267849 |
| maximal quotient of | #72267812 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387);