Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486091 |
| 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 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1049 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1049 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)(3,12)(4,7)(5,10)(6,11)(8,13)(9,16)(14,25)(15,21)(17,20)(18,24)(19,27)(22,26)(23,29)(28,31)(30,32);(2,7)(3,13)(4,16)(8,21)(9,10)(11,20)(12,19)(14,26)(17,29)(22,31)(23,24)(25,30);(3,14)(4,17)(8,22)(12,25)(13,26)(16,29)(19,30)(21,31);(1,3,6,14)(2,8,11,22)(4,15,17,28)(5,13,18,26)(7,19,20,30)(9,21,23,31)(10,12,24,25)(16,27,29,32)Polycyclic presentation:
2^9|d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72278573, #72338725, #72338739, … |
| maximal normal subgroup of | #72278573, #72338725, #72338739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393288);