Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 980 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,14,16,28)(4,18,17,9)(5,20,23,10)(6,13)(7,12)(8,15,26,21)(11,22)(19,25,32,27)(24,29,31,30);(1,3,7,4)(2,8)(5,21,23,32)(6,17,22,16)(9,10)(11,25)(12,27)(13,26)(14,30)(15,31,19,24)(18,29)(20,28);(1,4)(2,9)(3,7)(5,19)(6,16)(8,20)(10,27)(11,18)(12,28)(13,14)(15,23)(17,22)(21,31)(24,32)(25,30)(26,29)Polycyclic presentation:
2^9|a^2=f,b^2=gi,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fghi,f^c=fhi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72237360, #72237397, #72237399 |
| maximal normal subgroup of | #72237360, #72237397, #72237399 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40335);