Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126701 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1735 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #7898 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #405 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,8,14,25,32,9,15,24,26,7,13,5,22)(3,11,17,28,4,20,16,10,19,27,21,29,18,31,23,30);(1,3,7,18,9,19,8,4)(2,10)(5,23,24,21,25,16,6,17)(11,26)(12,27)(13,28)(14,29)(15,30)(20,22)(31,32);(1,4,8,19,9,18,7,3)(2,11,13,31,15,27,14,20)(5,21,6,23,25,17,24,16)(10,26,28,32,30,12,29,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224955, #72224957, #72224958, … |
| maximal normal subgroup of | #72224955, #72224957, #72224958, … |
| maximal quotient of | C2×#126873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34069);