Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119556 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1277 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1277 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,12,15,7)(4,17,19,25)(5,9)(6,10)(8,18,26,31)(11,20)(13,27,29,32)(14,23,16,22)(21,28,24,30);(1,3)(2,7)(4,13)(5,15)(6,16)(8,24)(9,12)(10,23)(11,22)(14,20)(17,21)(18,30)(19,28)(25,32)(26,27)(29,31);(1,4)(2,8,9,25)(3,13)(5,18)(6,19)(7,21,12,27)(10,26,11,17)(14,29)(15,30)(16,28)(20,31)(22,24,23,32)Polycyclic presentation:
2^9|a^2=g,c^2=d,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,g^b=gh,e^c=eh,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72211163, #72211186, #72228238, … |
| maximal normal subgroup of | #72211163, #72211186, #72228238, … |
| maximal quotient of | #72235134, #72235135, #72235141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26752);