Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141751 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1957 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 420 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10)(11,19,22,32)(12,24)(13,16)(14,28)(15,21,29,25)(17,20)(18,27)(23,38,37,33)(26,39)(30,34,40,36)(31,35);(2,6)(5,8)(9,11,20,34,27,37,12,25,13,22,35,36,14,23,26,21)(10,15,24,38,28,40,17,32,16,29,39,33,18,30,31,19);(1,2,3,5,4,6,7,8)(10,16)(15,29)(17,31)(18,28)(19,32)(24,39)(30,40)(33,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224821, #72224824, #72224825, … |
| maximal normal subgroup of | #72224821, #72224824, #72224825, … |
| maximal quotient of | #72225302, #72225303 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48948);