Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#107 |
| prime radical | C23 | #427 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 166 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:
(9,10,14,19,15,20,13,18,16,21,30,37,29,36,12,28)(11,22,25,17,26,33,31,34,27,38,23,35,32,40,24,39);(1,2,3,5,4,6,7,8)(9,11)(10,17)(12,23)(13,25)(14,31)(15,32)(16,27)(18,33)(19,38)(20,39)(21,35)(22,37)(24,30)(26,29)(28,40)(34,36);(9,12,29,30,16,13,15,14)(10,18,36,19,21,28,20,37)(11,23,26,24,27,25,32,31)(17,33,34,38,35,40,39,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^b=dg,e^b=eg,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72223649, #72223650, #72223651, … |
| maximal normal subgroup of | #72223649, #72223650, #72223651, … |
| maximal quotient of | C2×#129096, #72225412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36292);