Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8965 |
| prime radical | #3025 | C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,11,22,32,6,12,21,31,23,33,45,54)(3,13,16,34,18,35,42,56,17,36,41,55,43,57,60,64)(4,9,14,25,20,30,38,48,19,29,37,49,44,53,58,62)(8,24,27,40,28,47,52,39,26,46,50,59,51,61,63,15);(1,3,14,24,7,18,38,47,6,17,37,46,23,43,58,61)(2,8,25,34,11,28,48,56,12,26,49,55,33,51,62,64)(4,15,5,16,20,40,22,42,19,39,21,41,44,59,45,60)(9,13,10,27,30,35,32,52,29,36,31,50,53,57,54,63);(4,19)(5,21)(15,39)(16,41)(20,44)(22,45)(40,59)(42,60)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^b=egi,d^c=dg,f^c=fg,e^d=ei,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72342043, #72342048, #72343161, … |
| maximal normal subgroup of | #72342048, #72343161, #72343176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44025);