Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152123 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #60817 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3508 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
695 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,12,15,7)(4,16)(5,10,21,9)(6,11)(8,18)(13,27,28,22)(14,24,20,23)(17,30,31,29)(19,25,32,26);(1,3)(2,7)(4,17)(5,20)(6,14)(8,22)(9,24)(10,12)(11,23)(13,18)(15,21)(16,29)(19,28)(25,30)(26,27)(31,32);(1,4,6,18)(2,8)(3,13,14,17)(5,19,21,32)(7,22)(9,25)(10,26)(11,16)(12,27)(15,28,20,31)(23,29)(24,30)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fgh,g^b=gi,e^c=ei,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72339825, #72339827, #72339835, … |
| maximal normal subgroup of | #72339825, #72339827, #72339835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59319);