Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124650 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1527 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #376 | 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 ▷ #376 ▷ 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)(3,11,13,26,14,23,18,32)(4,15,16,30)(5,9)(6,10)(7,12,20,28,22,27,21,29)(8,17,24,31)(19,25);(1,3)(2,7)(4,12)(5,18)(6,14)(8,23)(9,20)(10,22)(11,24)(13,19)(15,26)(16,27)(17,28)(21,25)(29,31)(30,32);(4,16)(8,24)(11,23)(12,27)(15,30)(17,31)(26,32)(28,29)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,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338377, #72338394, #72338396 |
| maximal normal subgroup of | #72338377, #72338394, #72338396 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31846);