Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
833 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 833 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,14,8,13,5,12)(3,16,19,10,18,28,17,22)(4,20,23,29,24,11,21,30)(6,27,26,31,9,15,25,32);(1,3,9,11)(2,10,15,4)(5,17,25,30)(6,20,8,18)(7,19,26,29)(12,28,32,21)(13,22,27,24)(14,16,31,23);(1,4,9,10)(2,11,15,3)(5,16,26,21)(6,22,8,24)(7,28,25,23)(12,17,31,29)(13,20,27,18)(14,19,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72233440, #72233443, #72233444, … |
| maximal normal subgroup of | #72233440, #72233443, #72233444, … |
| maximal quotient of | #72317957, #72318033, #72318213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47044);