Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #489691 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19255 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 940 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(2,6)(3,4)(7,8)(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17);(2,7)(3,4)(6,8);(2,8)(6,7)Polycyclic presentation:
2^9|b^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72342556, SD32×AGL(1,ℤ/8), #72342600 |
| maximal normal subgroup of | #72342556, SD32×AGL(1,ℤ/8), #72342600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396888);