Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1790 |
| prime radical | #2679 | C4 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
772 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,25,26,32)(20,28,30,23)(21,24)(22,27)(29,31);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,26)(18,22,24,31)(20,29,30,27)(23,32,28,25);(3,9)(6,12)(10,15)(13,16)(19,26)(22,31)(23,28)(27,29)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=chi,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338680, #72338682, #72338685, … |
| maximal normal subgroup of | #72338680, #72338682, #72338685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36841);