Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485597 |
| 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 | C26 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1261 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 1261 conjugacy classes of subgroups — too many to draw. Only its 122 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:
(1,2,5,9)(3,11,12,22)(4,15,18,8)(6,10,19,24)(7,16,20,13)(14,26,27,32)(17,25,28,23)(21,30,31,29);(1,3,6,14)(2,7,10,21)(4,16,17,30)(5,12,19,27)(8,22,23,32)(9,20,24,31)(11,25,26,15)(13,28,29,18);(8,23)(11,26)(15,25)(22,32);(3,12)(7,20)(11,22)(13,16)(14,27)(21,31)(26,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209878, #72285450, #72285789, … |
| maximal normal subgroup of | #72285450, #72285789, #72285796, … |
| maximal quotient of | #72209872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392874);