Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141233 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×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×#493 | C22 |
| 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 | |
398 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 398 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)(9,10,15,19,14,24,12,22,17,20,13,18,16,11,21,23);(1,2,4,7,6,8,5,3)(9,11,17,24)(10,14,20,16)(12,22,21,23)(13,19,15,18);(1,3,5,8,6,7,4,2)(9,12,16,15,17,21,14,13)(10,18,24,23,20,19,11,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,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,d^b=dh,f^b=fh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72309675, #72309688, #72309700, … |
| maximal normal subgroup of | #72309675, #72309688, #72309700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48431);