Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D4 | SD16 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,12,17,14,18,13,23)(11,19,21,24,22,15,20,16);(2,6)(3,4)(7,8)(10,15,18,19)(11,20)(12,13)(16,17,24,23)(21,22);(2,7)(3,4)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214781, #72227119, #72227162, … |
| maximal normal subgroup of | #72214781, #72227119, #72227162, … |
| maximal quotient of | #72214625, #72225151, #72225154 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43563);