Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#1595 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1101 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 32 points:
(1,2)(3,8)(4,7)(5,10)(6,11)(9,15)(12,21)(13,19)(14,23)(16,30)(17,20)(18,25)(22,26)(24,29)(27,31)(28,32);(2,7)(4,15)(8,19)(9,10)(11,20)(13,26)(16,28)(17,29)(21,22)(23,30)(24,25)(31,32);(1,3,6,14)(2,8,11,23)(4,16,17,13)(5,12,18,27)(7,19,20,30)(9,22,24,32)(10,21,25,31)(15,28,29,26);(3,12)(8,21)(13,26)(14,27)(16,28)(19,22)(23,31)(30,32)Polycyclic presentation:
2^9|c^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72292039 |
| maximal normal subgroup of | #72292039 |
| maximal quotient of | #72292044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393985);