Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #12958 |
| prime radical | #3018 | C22 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,7,13,8,15,24,14)(3,10,17,29,18,30,32,26)(4,19,20,28,21,27,16,9)(5,11,22,25,6,12,23,31);(1,3,7,17,8,18,24,32)(2,9,13,19,15,28,14,27)(4,6,20,23,21,5,16,22)(10,25,29,12,30,31,26,11);(1,4)(2,10)(3,6)(5,18)(7,20)(8,21)(9,25)(11,27)(12,19)(13,29)(14,26)(15,30)(16,24)(17,23)(22,32)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,f^2=i,g^2=i,b^a=bd,c^a=ce,g^a=gh,e^b=eghi,h^b=hi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59654);