Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124865 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1525 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #376 | C23 |
| socle | C2 | #7631 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ C2×#376 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
462 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,11,13,26,14,20,18,32)(4,15,16,30)(5,9)(6,10)(7,12,21,27,23,29,22,28)(8,17,24,31)(19,25);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,21)(10,23)(11,24)(13,19)(15,26)(16,29)(17,27)(22,25)(28,31)(30,32);(1,4)(2,8,10,24)(3,12)(5,17)(6,16)(7,20,23,11)(9,15,25,30)(13,28)(14,29)(18,27)(19,31)(21,26,22,32)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=cfh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#124865, #72217950, #72227436 |
| maximal normal subgroup of | C2×#124865, #72217950, #72227436 |
| maximal quotient of | #72213050, #72213056, #72213057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32061);