Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12695 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #60725 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3508 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
621 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 76 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)(4,16)(5,10,20,9)(6,11)(8,17)(13,26,28,21)(14,23,19,22)(18,25,30,24)(27,32,31,29);(1,3)(2,7)(4,13)(5,19)(6,14)(8,21)(9,23)(10,12)(11,22)(15,20)(16,29)(17,27)(18,31)(24,26)(25,32)(28,30);(1,4,6,17)(2,8)(3,13,14,27)(5,18,20,30)(7,21)(9,25)(10,24)(11,16)(12,26)(15,28,19,31)(22,29)(23,32)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,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=fgh,g^b=gi,d^c=di,e^c=ei,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59287);