Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152670 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #23872 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
617 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 617 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11)(3,7,13,24)(4,12)(5,10,21,28)(8,20)(9,17)(14,25,16,26)(15,23,30,32)(18,27)(19,29,31,22);(1,3)(2,7,9,24,11,26,12,25)(4,13)(5,19)(6,16)(8,22,28,23,27,32,10,29)(14,17)(15,21)(18,31)(20,30);(1,4,6,17)(2,8,11,27)(3,13,16,14)(5,20,21,18)(7,22,26,32)(9,28,12,10)(15,31,19,30)(23,25,29,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gh,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72339871 |
| maximal normal subgroup of | #72339871 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59867);