Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245147 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2462 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 2462 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)(4,7)(6,8)(10,13)(11,15)(14,16);(3,8)(5,6)(11,15)(13,16);(2,5)(3,8)(6,7)(9,10)(11,16)(12,14)(13,15);(11,15)(13,16)Polycyclic presentation:
2^9|f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fi,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72209658, #72247548, #72247552, … |
| maximal normal subgroup of | #72247548, #72247552, #72247558, … |
| maximal quotient of | #72209867, #72228197, #72338262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152343);