Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130786 |
| 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 | #61089 | 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 | |
859 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 859 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:
(1,2,6,13,8,14,17,32,7,15,25,30,5,12,21,26)(3,16,19,9,18,27,23,28,20,31,22,29,24,10,4,11);(2,9)(4,19)(5,8)(6,21)(10,26)(11,14)(12,28)(13,31)(15,29)(16,30)(17,25)(18,24)(22,23)(27,32);(1,3)(2,10)(4,21)(5,24)(6,19)(7,20)(8,18)(9,26)(11,30)(12,31)(13,28)(14,16)(15,27)(17,23)(22,25)(29,32)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,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,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231612, #72231714, #72231719, … |
| maximal normal subgroup of | #72231612, #72231714, #72231719, … |
| maximal quotient of | #72250432, #72257044, C2×#130786 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37982);