Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
680 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 680 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(6,11,12,14)(13,20,19,18);(5,6,8,13,10,14,17,20,9,12,16,19,7,11,15,18);(1,3,4,2)(6,11,12,14)(8,16)(13,20,19,18)(15,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#128374, #72217920, #72219608, … |
| maximal normal subgroup of | C2×#128374, #72217920, #72219608, … |
| maximal quotient of | #72216852, C2×#128374 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35570);