Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152085 |
| 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 | #60714 | 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 | |
648 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 648 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)(3,12,14,7)(4,16)(5,10,20,9)(6,11)(8,17)(13,27,29,30)(15,22,19,23)(18,24,31,25)(21,28,26,32);(1,3)(2,7)(4,13)(5,19)(6,15)(8,21)(9,22)(10,12)(11,23)(14,20)(16,30)(17,28)(18,32)(24,27)(25,26)(29,31);(1,4,6,17)(2,8)(3,13,15,28)(5,18,20,31)(7,21)(9,24)(10,25)(11,16)(12,26)(14,29,19,32)(22,27)(23,30)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,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59281);