Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152566 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13041 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #33559 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
569 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 569 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,3)(4,15,6,10)(7,11,16,14)(8,9,12,13);(1,3,12,15,7,14,6,9)(2,8,10,16,11,4,13,5);(1,2)(3,8)(4,14)(5,9)(6,13)(7,11)(10,12)(15,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72213368, #72286053, #72286104, … |
| maximal normal subgroup of | #72213368, #72286053, #72286104, … |
| maximal quotient of | #72213302, #72241274, #72338944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59763);