Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#346 |
| prime radical | C23 | #346 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(10,15,16,25)(11,19)(17,30,28,26)(18,27,29,36)(20,32)(21,33)(31,39,38,37)(34,40);(9,10,11,17,12,18,20,31,13,15,21,26,22,27,34,37,14,16,19,28,23,29,32,38,24,25,33,30,35,36,40,39);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bh,c^a=cf,e^a=ei,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231855, #72231860, #72231861, … |
| maximal normal subgroup of | #72231855, #72231860, #72231861, … |
| maximal quotient of | #72322700, #72322703 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56704);