Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1962 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1962 conjugacy classes of subgroups — too many to draw. Only its 146 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)(7,14,16,10)(8,12)(9,13)(11,15,20,18)(17,19);(1,2)(3,4)(6,10)(8,17)(11,19)(12,20)(13,14)(15,18);(2,4)(5,7)(6,11)(8,18)(9,16)(10,12)(13,20)(14,19)(15,17)Polycyclic presentation:
2^9|a^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gh,e^c=ei,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215601, #72231756, #72233902, … |
| maximal normal subgroup of | #72215601, #72231756, #72233902, … |
| maximal quotient of | #72215559, #72231210, #72231381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54773);