Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137149 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 437 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)(3,10,14,16)(4,13,15,7)(5,8)(6,9)(11,12)(17,18,19,24,20,25,27,22,21,26,28,23,29,31,32,30);(1,3,11,13,6,14,5,7)(2,4,12,10,9,15,8,16)(18,22,26,30)(19,27)(20,29)(23,25,24,31)(28,32);(1,4)(2,7)(3,12)(5,16)(6,15)(8,14)(9,13)(10,11)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285149, #72285170, #72285180 |
| maximal normal subgroup of | #72285149, #72285170, #72285180 |
| maximal quotient of | #72343090, #72343092, #72343095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44457);