Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1774 conjugacy classes of subgroups — too many to draw. Only its 186 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:
(5,6,9,13,10,14,8,11)(7,15,16,20,18,19,17,12);(1,2)(3,4)(6,11,14,13)(12,19,20,15);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,15)(13,19)(14,20);(6,13)(7,16)(8,9)(11,14)(12,20)(17,18)Polycyclic presentation:
2^9|a^2=f,b^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72286950, #72287020, #72287040, … |
| maximal normal subgroup of | #72286950, #72287020, #72287040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384219);