Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1351 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1351 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,12,15,21)(4,8,17,23)(5,9)(6,11)(7,14,22,31)(10,19)(13,27,29,20)(16,25,18,24)(26,28,32,30);(1,3)(2,7,11,21)(4,13)(5,15)(6,14)(8,20,24,27)(9,22,10,12)(16,30)(17,28)(18,29)(19,31)(23,26,25,32);(1,4)(2,8,10,23)(3,13)(5,16)(6,18)(7,20,12,26)(9,25,11,24)(14,29)(15,30)(17,19)(21,27,22,32)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fhi62 conjugacy classes — view the full character table.
| Fitting subgroup of | #72235408 |
| maximal subgroup of | #72211186, #72228237, #72228238, … |
| maximal normal subgroup of | #72211186, #72228237, #72228238, … |
| maximal quotient of | #72235132, #72235140, #72235225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17376);