Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(13,14,17,23,18,20,26,27,19,15,21,24,16,22,25,28);(2,4)(5,6,7,9,8,10,11,12)(14,20,15,22)(17,21)(23,28,24,27)(25,26);(13,15,17,24,18,22,26,28,19,14,21,23,16,20,25,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=cg,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1313, #72227128, #72227150, … |
| maximal normal subgroup of | D4×#1313, #72227128, #72227150, … |
| maximal quotient of | #72225537, #72225540, C2×#128255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35451);