Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62245 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1221 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1221 conjugacy classes of subgroups — too many to draw. Only its 182 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 — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,23)(17,24,19,21);(1,2,6,8,4,7,3,5)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24);(9,12)(10,15)(11,17,18,19)(13,20)(14,21,22,24)(16,23);(1,2,6,8,4,7,3,5)Polycyclic presentation:
2^9|a^2=i,b^2=eh,c^2=f,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72343104, D8×#458, #72343248 |
| maximal normal subgroup of | #72343104, D8×#458, #72343248 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209071);