Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141213 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #469 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10,15,20,17,21,16,11)(12,14,22,24,19,13,23,18);(2,6)(3,4)(7,8)(9,11,14,22,16,10,18,19,17,20,13,23,15,21,24,12);(9,12,14,20,16,22,18,21,17,19,13,11,15,23,24,10)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,d^2=ghi,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72342467, #72342469, #72342472, … |
| maximal normal subgroup of | #72342467, #72342469, #72342472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48477);