Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 385 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,4,7)(3,5,8,6)(10,14,16,21)(11,18)(12,19)(15,23,22,17)(20,24);(2,5,7,6)(9,10,11,14,12,17,18,22,13,16,20,21,19,23,24,15);(9,11,19,24,13,20,12,18)(10,15,17,14,16,22,23,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=egi,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=ch,e^b=ei,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319759, #72319765, #72319839, … |
| maximal normal subgroup of | #72319759, #72319765, #72319839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35806);