Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#322 |
| prime radical | C23 | #322 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
208 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(1,2,3,7,4,9,11,15,5,8,10,16,12,6,13,14);(2,6,8,9)(3,10)(7,15,16,14)(11,13)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(18,22)(23,28)(24,29)(30,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bf,e^a=eh,c^b=ci,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72227018, #72231871, #72231872, … |
| maximal normal subgroup of | #72227018, #72231871, #72231872, … |
| maximal quotient of | #72231871, C2×#122219, #72231880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29415);