Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1389 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1389 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,14,20,23)(12,16,21,15)(13,17)(18,24)(19,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14,17,23)(12,18,21,19)(13,20)(15,22)(16,24);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18,20,19)(13,21)(14,22,23,24)(16,17);(2,5)(3,7)(9,12)(10,16,17,15)(11,19)(13,21)(14,22,23,24)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ef,d^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72318408, #72318425, #72318453, … |
| maximal normal subgroup of | #72318408, #72318425, #72318453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387025);