Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152089 |
| 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 | #532 | C23 |
| prime radical | #60704 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3455 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 614 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:
(1,2,5,11)(3,13,16,30)(4,10,18,27)(6,8,19,23)(7,12,21,29)(9,24,25,17)(14,22,20,32)(15,26,31,28);(2,8)(4,6)(5,19)(9,22)(10,12)(13,26)(15,17)(16,24)(18,21)(20,31)(23,27)(30,32);(1,3,7,14)(2,9)(4,15,6,17)(5,20,21,16)(8,22)(10,13)(11,28)(12,26)(18,24,19,31)(23,30)(25,29)(27,32)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,e^2=i,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,f^c=fi,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72292696, C2×#152089 |
| maximal normal subgroup of | #72292696, C2×#152089 |
| maximal quotient of | C2×#152089, #72293065, #72293070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59285);