Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #146230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2094 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(9,10,13,23,16,11,12,17,15,20,24,18,14,19,22,21);(1,2,4,7)(3,5,8,6)(9,11,14,20,15,19,16,10)(12,17,13,18,22,21,24,23);(2,5)(3,8)(6,7)(9,11)(10,16)(12,17)(13,18)(14,20)(15,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=dg,b^2=degh,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224921, #72224924, #72224925 |
| maximal normal subgroup of | #72224921, #72224924, #72224925 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53467);