Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149901 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| 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 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
604 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 604 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2)(3,4)(5,6,8,13,10,15,22,32)(7,16,18,33,17,31,34,29)(9,14,24,12,25,26,21,30)(11,19,28,35,27,23,36,20);(5,7)(6,11,26,33,15,27,14,29)(8,20)(9,23)(10,17)(12,28,32,16,30,36,13,31)(18,21)(19,25)(22,35)(24,34);(2,4)(7,17)(11,27)(16,31)(18,34)(19,23)(20,35)(28,36)(29,33)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72279392 |
| maximal normal subgroup of | #72279392 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59852);