Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184792 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#127 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
6646 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 6646 conjugacy classes of subgroups — too many to draw. Only its 410 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)(7,10)(8,9)(11,12)(14,17)(15,19);(6,9)(10,12)(14,17)(15,19);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14,15,18,16,17,19,20);(7,11)(10,12)Polycyclic presentation:
2^9|c^2=h,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225881, #72226077, #72226354, … |
| maximal normal subgroup of | #72225881, #72226077, #72226354, … |
| maximal quotient of | #72225875, C2×#382394, #72318390 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289590);