Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7533 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1409 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #494 | C23 |
| socle | C2 | C2×#1409 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,5,12,8,14,3,10)(4,11,19,30,21,26,32,24)(6,13,16,29,7,23,17,31)(9,22,25,18,27,15,28,20);(2,9,14,27)(4,18)(6,7)(10,24,12,30)(11,23,26,13)(15,19)(16,17)(20,21)(22,32)(25,29,28,31);(1,3,8,5)(2,10,14,12)(4,15,21,22)(6,16,7,17)(9,24,27,30)(11,28,26,25)(13,29,23,31)(18,19,20,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,d^2=gh,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=dgh,e^b=efg,f^b=fhi,g^b=gi,e^d=egh,f^e=fhi,g^e=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72283179, #72283253, #72283288, … |
| maximal normal subgroup of | #72283179, #72283253, #72283288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60724);