Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251827 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1910 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2,4,6)(3,5,7,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,23,24,22);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15)(13,18,21,16)(17,20)(19,22)(23,24);(5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)Polycyclic presentation:
2^9|a^2=hi,b^2=eh,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213228, #72213781, #72213791, … |
| maximal normal subgroup of | #72213228, #72213781, #72213791, … |
| maximal quotient of | #72233149, #72233178, #72235018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159051);