Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152071 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12694 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,11,14,27)(4,15,17,31)(5,19,21,9)(6,10)(7,12,23,29)(8,18)(13,26,20,28)(16,24,30,22)(25,32);(1,3)(2,7,10,22)(4,16)(5,20)(6,13)(8,11,25,26)(9,23,19,24)(12,17)(14,21)(15,27,31,28)(18,30)(29,32);(4,17)(8,25)(11,26)(12,16)(15,31)(18,32)(27,28)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=ei,f^b=fghi,g^b=gi,d^c=di,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213429, #72216507, #72216516, … |
| maximal normal subgroup of | #72213429, #72216507, #72216516, … |
| maximal quotient of | C2×#152071, C3×#152071 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59267);