Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18186 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 683 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 32 points:
(1,2)(3,14)(4,10)(5,8)(6,23)(7,13)(9,16)(11,21)(12,20)(15,30)(17,28)(18,27)(19,25)(22,24)(26,32)(29,31);(2,8)(4,18)(5,21)(6,20)(9,10)(11,12)(13,25)(14,30)(15,31)(17,22)(19,32)(23,26)(24,29)(27,28);(1,3,7,16)(2,9)(4,19,15,5)(6,17,20,22)(8,10)(11,27)(12,28)(13,14)(18,32,31,21)(23,24)(25,30)(26,29);(1,4,20,31,7,15,6,18)(2,10,12,29,13,30,23,27)(3,5,22,32,16,19,17,21)(8,24,26,9,25,28,11,14)Polycyclic presentation:
2^9|c^2=h,d^2=gh,e^2=gi,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,e^d=ei,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231715, C2×#486086 |
| maximal normal subgroup of | #72231715, C2×#486086 |
| maximal quotient of | #72250610, #72250961, C2×#486086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393282);