Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453974 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2126 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 2126 conjugacy classes of subgroups — too many to draw. Only its 218 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:
(2,3)(5,6)(7,13)(8,16)(9,12)(10,14)(11,17)(15,20)(18,19);(1,2,4,3)(8,17)(10,19)(11,16)(14,18);(2,3)(5,7)(6,10)(8,18)(9,15)(11,20)(12,19)(13,16)(14,17);(1,3,4,2)(8,17)(11,16)(13,20)(14,18)Polycyclic presentation:
2^9|b^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312508, #72312585, #72312590 |
| maximal normal subgroup of | #72312508, #72312585, #72312590 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361552);