Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1799 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20)(12,24)(13,18,27,16)(14,19)(15,22)(17,26)(21,34)(23,33,36,40)(25,31,38,32)(28,39,30,37)(29,35);(9,11,12,22,14,21,26,35)(10,15,17,20,19,29,24,34)(13,23,25,37,27,36,38,39)(16,30,31,40,18,28,32,33);(10,16)(11,21)(13,27)(15,28)(17,31)(18,19)(20,33)(22,35)(24,32)(25,38)(29,30)(34,40)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ci,e^c=ei,f^c=fgi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264850, #72264859, #72307992 |
| maximal normal subgroup of | #72264850, #72264859, #72307992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41433);