Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135783 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1844 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #492 |
| 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 | |
262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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)(11,19,20,31)(12,15)(13,25)(14,27)(16,21,28,33)(17,26)(18,24)(22,36,34,32)(23,37)(29,38)(30,40,39,35);(1,2,3,5,4,6,7,8)(10,15)(16,28)(17,29)(18,27)(19,31)(25,37)(30,39)(32,36);(2,6)(5,8)(9,11,13,21,14,22,26,35,12,20,23,33,24,34,38,40)(10,16,17,19,18,30,25,32,15,28,29,31,27,39,37,36)Polycyclic presentation:
2^9|a^2=f,b^2=di,c^2=e,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309711, #72309735, #72309740, … |
| maximal normal subgroup of | #72309711, #72309735, #72309740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42980);