Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C23 | #428 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,12,18,13,19,22,17,14,20,24,15,23,16,11,21);(1,2,3,6,4,5,7,8)(10,15)(11,12)(13,23)(16,21)(17,19)(18,20)(22,24);(2,5)(6,8)(10,16,20,19)(11,22)(12,24)(15,21,18,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=gi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=cg,d^a=di,c^b=ch,d^b=dg,e^b=egi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225334, #72227172, #72227176, … |
| maximal normal subgroup of | #72225334, #72227172, #72227176, … |
| maximal quotient of | #72225332, #72227236 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36320);