Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
721 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 721 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(1,2,6,11)(3,13,16,26)(4,17,22,9)(5,10,7,12)(8,18,25,14)(15,30,23,32)(19,20,27,21)(24,29,28,31);(1,3)(2,8)(4,18)(5,23)(6,16)(7,15)(9,26)(10,28)(11,25)(12,24)(13,17)(14,22)(19,30)(20,31)(21,29)(27,32);(1,2,7,12)(3,8,15,24)(4,19,21,17)(5,10,6,11)(9,22,27,20)(13,29,30,18)(14,26,31,32)(16,25,23,28)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72235677, #72235690, #72235698, … |
| maximal normal subgroup of | #72235677, #72235690, #72235698, … |
| maximal quotient of | #72237737, #72237858, #72284155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49765);