Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
970 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 970 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,7,9,11,12,10)(14,17)(15,18)(19,20);(1,3,4,2)(6,10)(7,11)(8,12);(13,14,18,20,16,19,15,17);(5,7,12,6,9,10,8,11)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh98 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D82, #72225198, #72225202, … |
| maximal normal subgroup of | C4×D82, #72225198, #72225202, … |
| maximal quotient of | #72216416, #72225406, #72226848 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138757);