Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #331 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#331 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
470 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(2,6,7,8)(9,10,11,16,12,15,19,23,13,17,20,24,21,14,18,22);(1,2)(3,6)(4,8)(5,7)(10,14)(11,18)(12,21)(15,17)(16,24)(19,20);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|a^2=e,c^2=fghi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216424, #72216426, #72216470, … |
| maximal normal subgroup of | #72216424, #72216426, #72216470, … |
| maximal quotient of | #72216566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31411);