Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7349 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1317 |
| commutator | C24 | C4×C8 |
| Frattini, cosocle | C2×#379 | C22 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ C2×#379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,8,12)(7,10,14,17,13,18,15,20)(9,11,16,19);(1,2,3,4)(5,7)(6,10,11,18)(8,15)(9,13)(12,17,19,20)(14,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=df,g^b=gi,d^c=dh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237482, #72238058, #72238404, … |
| maximal normal subgroup of | #72237482, #72238058, #72238404, … |
| maximal quotient of | #72238066 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1268);