Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133924 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
813 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 813 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,4,6)(5,8)(9,10)(11,17)(12,16,21,24)(13,22)(14,18)(15,19,23,20);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,18,23)(12,20,21,19)(16,22,24,17);(1,4,5,3)(2,7,8,6)(9,11,14,13)(12,19,21,20)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,f^b=fgi,f^c=fhi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233271, #72237394 |
| maximal normal subgroup of | #72233271, #72237394 |
| maximal quotient of | #72233268, #72237357 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41182);