Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94533 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7344 |
| commutator | C4⋊Q8 | C2×C8 |
| Frattini, cosocle | #1963 | C22 |
| prime radical | #3549 | C4 |
| LCS | G ▷ C4⋊Q8 ▷ 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:
(1,2,4,10,5,12,18,28,7,11,20,29,6,13,21,30)(3,8,15,22,17,23,31,32,14,9,24,25,16,26,19,27);(2,8)(3,14)(4,18)(5,6)(9,11)(10,25)(12,26)(13,23)(15,31)(19,24)(20,21)(22,29)(27,30)(28,32)Polycyclic presentation:
2^9|a^2=d,c^2=i,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,c^b=ci,d^b=dehi,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 | #72339715, #72339795, #72339797, … |
| maximal normal subgroup of | #72339715, #72339795, #72339797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1732);