Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,13,10,15,23,33)(7,16,19,27,17,29,35,11)(9,14,25,12,26,30,22,31)(18,34,24,32,21,28,20,36);(1,2,3,4)(5,7)(6,11)(8,21)(9,24)(10,17)(12,29)(13,32)(14,34)(15,27)(16,31)(18,23)(19,22)(20,26)(25,35)(28,30)(33,36);(7,17)(11,27)(16,29)(18,21)(19,35)(20,24)(28,34)(32,36)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bd,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^d=ehi,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279340, #72279392 |
| maximal normal subgroup of | #72279340, #72279392 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59005);