Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C23 | #441 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2,4,7,5,3,6,8)(9,10,13,20,15,21,29,19,16,22,30,36,14,18,12,28)(11,23,25,34,26,35,39,38,27,37,40,17,31,33,24,32);(9,11,14,26,16,27,15,31)(10,17,18,32,22,34,21,38)(12,25,30,39,29,40,13,24)(19,35,20,37,28,33,36,23);(1,3,4,8,5,2,6,7)(10,18,22,21)(12,29)(13,30)(17,32,34,38)(19,36,28,20)(23,33,37,35)(24,39)(25,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216219, #72216451, #72216456, … |
| maximal normal subgroup of | #72216219, #72216451, #72216456, … |
| maximal quotient of | #72223570, C2×#129166, #72225332 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36362);