Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105838 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2340 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 2340 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,16,14,18,12,17)(11,19,22,24,20,15,21,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,18,19)(12,21)(13,22)(14,20)(16,23,17,24);(10,16)(12,13)(15,23)(17,18)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235080, #72238125, #72238126, … |
| maximal normal subgroup of | #72235080, #72238125, #72238126, … |
| maximal quotient of | #72238663, #72238685, #72261434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13045);