Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #487635 |
| 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 | #56346 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 1038 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:
(1,2,5,10)(3,12,14,8)(4,7,16,9)(6,11,18,25)(13,27,28,23)(15,22,30,19)(17,20,32,24)(21,29,26,31);(2,7)(3,13)(4,16)(8,21)(9,10)(11,20)(12,26)(14,28)(15,31)(17,32)(19,23)(22,27)(24,25)(29,30);(3,14)(4,17)(8,12)(13,28)(15,31)(16,32)(19,22)(21,26)(23,27)(29,30);(1,3,5,14)(2,8,10,12)(4,15,16,30)(6,13,18,28)(7,19,9,22)(11,23,25,27)(17,29,32,31)(20,21,24,26)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233399 |
| maximal normal subgroup of | #72233399 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395175);