Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8375 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61063 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
832 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 832 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,6,12,8,13,3,15,7,14,16,29,4,10,18,28)(5,11,20,9,21,26,32,24,22,30,23,25,19,27,17,31);(1,3)(2,9)(4,16)(5,20)(6,8)(7,18)(10,24)(11,29)(12,30)(13,31)(14,25)(15,26)(17,21)(19,32)(22,23)(27,28);(1,4,7,8)(2,10,14,13)(3,16,18,6)(5,21,22,19)(9,24,25,31)(11,26,30,27)(12,28,29,15)(17,23,32,20)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,f^c=fg,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231603, #72231700, #72231719, … |
| maximal normal subgroup of | #72231603, #72231700, #72231719, … |
| maximal quotient of | #72250389, #72257031, #72280623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37981);