Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2968 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 2968 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,13)(8,16)(9,12)(10,15)(11,18)(14,20)(17,19);(10,19)(13,20);(1,3,4,2)(5,7)(6,10)(8,17)(9,14)(11,20)(12,19)(13,16)(15,18);(8,18)(11,16)(13,20)(15,17)Polycyclic presentation:
2^9|a^2=h,c^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216097, #72219640, #72219647, … |
| maximal normal subgroup of | #72216097, #72219640, #72219647, … |
| maximal quotient of | #72242954, #72257943 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280836);