Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245225 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3531 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 3531 conjugacy classes of subgroups — too many to draw. Only its 156 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:
(1,2)(3,5)(4,6)(7,8)(13,16);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16);(5,8)(9,11)(10,13)(12,15)(14,16);(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^9|b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72209653, #72209767, #72210439, … |
| maximal normal subgroup of | #72209767, #72210439, #72210565, … |
| maximal quotient of | B(5,𝔽2), #72209835, #72209840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152421);