Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2710 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 2710 conjugacy classes of subgroups — too many to draw. Only its 346 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,13,9,12,17,7)(8,11,16,14,15,19,18,20);(2,4)(7,13)(10,17)(14,20)(16,18);(5,7,10,12,9,13,17,6)(8,14,18,19,15,20,16,11);(2,4)(5,8)(6,11)(7,14)(9,15)(10,18)(12,19)(13,20)(16,17)Polycyclic presentation:
2^9|a^2=h,c^2=fh,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72210524, #72210563, #72215938, … |
| maximal normal subgroup of | #72210524, #72210563, #72215938, … |
| maximal quotient of | C2×#374009 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281205);