Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133080 |
| 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×#2661 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1008 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)(3,13,16,8)(4,17,19,7)(5,10)(6,12)(9,18,22,32)(11,20)(14,25,29,26)(15,27,30,21)(23,28,24,31);(2,7)(3,14)(4,18)(8,23)(9,10)(11,22)(12,17)(13,27)(15,30)(16,29)(19,32)(21,26)(24,25)(28,31);(1,3)(2,8)(4,15)(5,16)(6,14)(7,21)(9,24)(10,13)(11,25)(12,26)(17,23)(18,31)(19,28)(20,29)(22,27)(30,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72238366, #72238367, #72238369, … |
| maximal normal subgroup of | #72238366, #72238367, #72238369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40276);