Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152136 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12707 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #536 | C23 |
| prime radical | #61648 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3485 ◁ G | |
| LECS | G ▷ #536 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,5,10,7,12,19,28)(3,9,14,25,6,11,20,29)(4,15,18,26,16,22,30,32)(8,21,24,17,23,31,27,13);(2,8)(3,6)(4,16)(5,14)(9,15)(10,26)(11,22)(12,23)(18,31)(19,20)(21,30)(24,29)(25,27)(28,32);(1,3,7,6)(2,9,12,11)(4,17,16,13)(5,14,19,20)(8,22,23,15)(10,25,28,29)(18,31,30,21)(24,32,27,26)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fhi,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72292692, C2×#152143 |
| maximal normal subgroup of | #72292692, C2×#152143 |
| maximal quotient of | C2×#152143, #72292955, #72292967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59339);