Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7527 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1408 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| socle | C2 | C2×#1408 |
| prime radical | #19616 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3147 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 303 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)(3,8)(4,15,18,7)(5,20,17,11)(6,10)(9,16,22,19)(12,26,29,21)(13,31,28,25)(14,24)(23,27,32,30);(2,7)(4,16)(5,17)(8,21)(9,20)(10,22)(11,15)(12,27)(13,28)(18,19)(23,31)(24,32)(25,26)(29,30);(1,3,6,14)(2,8,10,24)(4,12,19,30)(5,13,17,28)(7,21,22,32)(9,23,15,26)(11,25,20,31)(16,27,18,29)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=ghi,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=efgi,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 | #72283514, #72283558, #72283572, … |
| maximal normal subgroup of | #72283514, #72283558, #72283572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60715);