Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94532 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7344 |
| commutator | #127 | C2×C8 |
| Frattini, cosocle | #1963 | C22 |
| prime radical | #3549 | C4 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1963 ◁ G | |
| LECS | G ▷ #1963 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 26 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 — 2 generators on 32 points:
(1,2,4,9,6,10,16,27,7,12,19,26,5,11,18,29)(3,8,13,21,14,22,30,25,15,23,31,24,20,28,17,32);(2,8)(4,16)(5,6)(9,24)(10,28)(11,22)(12,23)(13,17)(14,20)(18,19)(21,26)(25,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=d,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=deh,e^b=eh,f^b=fi,g^b=gh,d^c=dghi,e^c=eh,g^c=gi,e^d=ehi,g^d=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72338328, #72338410, #72338411 |
| maximal normal subgroup of | #72338328, #72338410, #72338411 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1730);