Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1982 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 474 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,14,18,24,20,12,22,13,16,23,21,15,11,19,17);(1,2,5,3,6,8,4,7)(9,11,15,10,13,20,24,16)(12,22,14,21,19,17,23,18);(1,3,5,8,6,7,4,2)(10,16)(11,20)(17,22)(18,21)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=ei,d^2=ghi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,c^b=cg,d^b=dh,f^b=fh,h^b=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216361, #72216593, #72225246, … |
| maximal normal subgroup of | #72216361, #72216593, #72225246, … |
| maximal quotient of | #72216299, #72227226 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48492);