Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9,11,12)(13,14,19,24,20,21,17,27,18,23,28,26,16,15,25,22);(5,6,10,9,8,11,7,12)(13,15,16,14,18,21,20,23)(17,27,19,26,25,22,28,24);(2,4)(13,16,18,20)(14,21,23,15)(17,19,25,28)(22,24,27,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318781, #72318785, #72318789, … |
| maximal normal subgroup of | #72318781, #72318785, #72318789, … |
| maximal quotient of | #72322404, #72322507 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43810);