Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #102 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
292 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,22,26,23,27,31,32,24,28,21,30,20,19,29,25);(17,19)(18,23)(20,28)(21,25)(22,32)(24,27)(26,29)(30,31);(17,20,24,23)(18,19,28,27)(21,31,22,29)(25,30,32,26)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286797, #72287052, #72287116, … |
| maximal normal subgroup of | #72286797, #72287052, #72287116, … |
| maximal quotient of | #72287202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29660);