Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #35272 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1342 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,12,15,20)(8,11,16,10)(9,13)(14,19)(17,18);(1,2)(3,4)(6,10)(8,16)(11,13)(12,18)(14,17)(19,20);(2,4)(5,7)(8,17)(9,15)(10,11)(14,16)(18,19)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=ghi,d^c=di,g^c=gh,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233902, #72234081, #72234096, … |
| maximal normal subgroup of | #72233902, #72234081, #72234096, … |
| maximal quotient of | #72233901, #72234067, #72234179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54930);