Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124661 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1545 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7606 |
| prime radical | #3449 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,12,14,22,15,21,18,7)(4,8)(5,9)(6,11)(10,19)(13,27,29,20,28,26,31,32)(16,24)(17,25)(23,30);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,21)(10,12)(11,22)(14,19)(16,28)(17,31)(23,26)(24,32)(25,27)(29,30);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,25)(10,23)(11,24)(12,26)(14,29)(15,28)(18,31)(19,30)(21,27)(22,32)Polycyclic presentation:
2^9|a^2=e,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31857);