Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1594 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 266 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,10)(7,12)(8,11)(14,17)(15,19)(18,20);(1,2)(3,4)(5,6,8,10,9,7,11,12);(2,4)(13,14,15,17,16,18,19,20);(5,7,8,12,9,6,11,10)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg98 conjugacy classes — view the full character table.
| maximal subgroup of | #72214773, #72219628 |
| maximal normal subgroup of | #72214773, #72219628 |
| maximal quotient of | #72214375, #72224930, #72224941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 205799);