Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #263291 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #18186 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
868 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 868 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(18,22)(19,25)(20,29)(24,31)(27,28)(30,32);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,18,20,22,21,24,29,31)(19,26,27,32,28,23,25,30);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19)(18,23)(20,27)(21,28)(22,30)(24,26)(25,29)(31,32);(17,18,20,22,21,24,29,31)(19,23,27,30,28,26,25,32)Polycyclic presentation:
2^9|b^2=ehi,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72305440, #72305453, #72305464, … |
| maximal normal subgroup of | #72305440, #72305453, #72305464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 170493);