Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,18,14,16,13,19)(11,15,20,24,21,23,22,17);(10,15,16,23)(11,20)(12,13)(17,18,24,19)(21,22);(2,5)(6,8)(10,16)(15,23)(17,24)(18,19)Polycyclic presentation:
2^9|a^2=e,b^2=fgi,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,g^b=gi,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265302, #72265439, #72265449, … |
| maximal normal subgroup of | #72265302, #72265439, #72265449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31888);