Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249319 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2567 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 2567 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 24 points:
(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(1,2)(3,5)(4,6)(7,8)(14,23)(17,24);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(1,2)(3,5)(4,6)(7,8)(12,22)(15,20)(17,24)(19,21)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72209767, #72209769, #72227597, … |
| maximal normal subgroup of | #72209767, #72227597, #72227707, … |
| maximal quotient of | #72209840, #72227564, #72232713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156527);