Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1595 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #9017 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 32 points:
(1,2,5,10,6,11,19,28,7,12,21,29,20,27,4,9)(3,13,15,23,16,26,14,30,17,31,18,8,22,24,32,25);(1,3,6,22,7,17,20,16)(2,8,11,30,12,23,27,25)(4,18,5,14,19,15,21,32)(9,26,10,13,28,24,29,31);(4,19)(5,21)(14,32)(15,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72217920, #72219620, #72224561, … |
| maximal normal subgroup of | #72217920, #72219620, #72224561, … |
| maximal quotient of | #72215054, #72215055, #72215063 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44352);