Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 793 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:
(1,2,6,10,5,11,22,19)(3,13,15,24,16,21,23,32)(4,18,14,28,20,9,29,17)(7,27,25,31,8,12,26,30);(1,3,14,11,7,28,32,12)(2,8,17,24,27,5,16,29)(4,19,6,23,21,30,25,18)(9,20,10,22,15,13,31,26);(4,20)(6,22)(9,18)(10,19)(13,21)(15,23)(25,26)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=h,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 | #72233615, #72233616, #72233618, … |
| maximal normal subgroup of | #72233615, #72233616, #72233618, … |
| maximal quotient of | #72237844, #72237896, #72284166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47102);