Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | #3486 | C4 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
603 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 603 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:
(1,2,5,12,7,11,21,26)(3,13,15,29,16,28,31,23)(4,17,18,27,6,9,20,10)(8,14,24,30,25,22,32,19);(1,3)(2,8,17,29,11,25,9,23)(4,14)(5,19)(6,22)(7,16)(10,24,26,13,27,32,12,28)(15,18)(20,31)(21,30);(1,4,7,6)(2,9,11,17)(3,14,16,22)(5,20,21,18)(8,23,25,29)(10,12,27,26)(13,24,28,32)(15,19,31,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,e^b=ei,f^b=fg,g^b=gh,d^c=di,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279401, #72283075 |
| maximal normal subgroup of | #72279401, #72283075 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59873);