Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142540 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #23948 | C2 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 700 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:
(1,2,7,23)(3,12,14,29)(4,18,24,21)(5,9,6,11)(8,25,26,13)(10,20,32,22)(15,19,27,30)(16,28,17,31);(1,3)(2,8)(4,19)(5,17)(6,16)(7,14)(9,15)(10,28)(11,27)(12,18)(13,22)(20,25)(21,29)(23,26)(24,30)(31,32);(2,9)(3,8)(4,20)(11,23)(12,13)(14,26)(15,17)(16,27)(19,28)(22,24)(25,29)(30,31)Polycyclic presentation:
2^9|a^2=gi,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72251832, #72251999, #72252002, … |
| maximal normal subgroup of | #72251832, #72251999, #72252002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49770);