Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #239359 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3186 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1382 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1382 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,24)(22,23);(1,3)(2,5)(4,7)(6,8)(10,15)(11,20)(12,13)(16,23)(17,18)(21,22);(5,8)(9,11,12,22,14,21,13,20)(10,16,17,24,18,23,15,19);(11,21)(16,23)(19,24)(20,22)Polycyclic presentation:
2^9|c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339559, #72339562, #72339567, … |
| maximal normal subgroup of | #72339559, #72339562, #72339567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146555);