Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138556 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19101 | C2 |
| 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 | |
402 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,7,5,8,4,6)(10,14,16,19)(11,18)(12,21)(15,20)(17,24,23,22);(2,6)(3,4)(7,8)(9,10,15,22,21,23,20,14,13,16,18,24,12,17,11,19);(2,7)(3,4)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ehi,f^c=fhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227032, #72227163, #72227176, … |
| maximal normal subgroup of | #72227032, #72227163, #72227176, … |
| maximal quotient of | #72227021, #72227245, #72227246 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45849);