Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232273 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1974 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1974 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(6,9)(7,12)(10,11)(13,14)(15,19)(16,18)(17,20);(2,4)(5,6)(7,11)(8,10)(9,12)(14,18)(15,16)(19,20);(6,10)(9,11)(13,15,17,16)(14,19,20,18)Polycyclic presentation:
2^9|d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249998, #72311641, #72311653, … |
| maximal normal subgroup of | #72249998, #72311641, #72311653, … |
| maximal quotient of | #72217673, #72217716, #72217725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139479);