Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2429 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 2429 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 32 points:
(1,2)(3,14)(4,8)(5,12)(6,11)(7,13)(9,16)(10,18)(15,29)(17,28)(19,30)(20,27)(21,26)(22,24)(23,32)(25,31);(2,8)(4,18)(9,23)(10,13)(11,27)(12,24)(14,29)(15,19)(20,22)(25,26)(28,30)(31,32);(1,3)(2,9)(4,19)(5,21)(6,16)(7,17)(8,23)(10,25)(11,28)(12,14)(13,26)(15,18)(20,31)(22,32)(24,29)(27,30);(1,3)(2,9)(4,15)(5,16)(6,21)(7,17)(8,23)(10,25)(11,14)(12,28)(13,26)(18,19)(20,31)(22,32)(24,30)(27,29)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72260481, #72304193, #72340414 |
| maximal normal subgroup of | #72260481, #72304193, #72340414 |
| maximal quotient of | #72261118, #72304195, #72324317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419027);