Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #18547 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,7,11,8,3,5,10)(4,9,15,24,17,13,31,32)(6,12,19,27,22,14,21,29)(16,25,23,28,20,30,18,26);(1,3,8,2)(4,13)(5,10,7,11)(6,14,22,12)(9,17)(15,24)(16,30)(18,26)(19,27,21,29)(20,25)(23,28)(31,32);(1,4)(2,9)(3,13)(5,18)(6,20)(7,23)(8,17)(10,26)(11,28)(12,30)(14,25)(15,19)(16,22)(21,31)(24,27)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=egh,f^b=fi,h^b=hi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72339821, #72339858, #72339859, … |
| maximal normal subgroup of | #72339821, #72339858, #72339859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59805);