Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347042 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3607 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 3607 conjugacy classes of subgroups — too many to draw. Only its 196 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,8)(13,16);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16);(1,3,4,7)(2,5,6,8);(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^9|c^2=h,d^2=i,b^a=be,c^a=cf,d^a=dg,f^b=fh,g^b=gi,e^c=eh,e^d=ei59 conjugacy classes — view the full character table.
| maximal subgroup of | #72210342, C3×#347047, #72215606, … |
| maximal normal subgroup of | #72210342, C3×#347047, #72215606, … |
| maximal quotient of | #72210559, C3×#347047, #72219235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254243);