Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107114 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C24 | C2×C42 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1514 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1514 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,11,14,19)(4,8)(5,9)(6,10)(7,13,20,29)(12,26,28,18)(15,22)(16,23)(17,24)(21,30)(25,32,31,27);(1,3)(2,7,10,19)(4,12)(5,14)(6,13)(8,18,22,31)(9,20,24,11)(15,27)(16,28)(17,29)(21,25,23,26)(30,32);(1,4,5,16)(2,8,9,23)(3,12,14,28)(6,15,17,30)(7,18,20,26)(10,22,24,21)(11,25,19,31)(13,27,29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235861, #72235864, #72235868, … |
| maximal normal subgroup of | #72235861, #72235864, #72235868, … |
| maximal quotient of | #72237932, #72237935 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14311);