Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144633 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10390 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 661 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,6)(4,8,13,7)(5,9)(10,15,12,14)(11,16);(1,3)(2,6)(4,12,13,10)(5,11)(7,14,8,15)(9,16);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)Polycyclic presentation:
2^9|a^2=d,b^2=d,e^2=i,b^a=bd,c^a=ce,e^a=egi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^c=ei,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72210018, #72216336, #72216370, … |
| maximal normal subgroup of | #72210018, #72216336, #72216370, … |
| maximal quotient of | #72210015, #72227556, #72235245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51829);