Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130780 |
| 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 | #61079 | 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 | |
835 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 835 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:
(1,2,7,13,9,14,3,16,8,15,19,28,6,12,20,32)(4,21,23,30,22,27,17,29,18,31,5,11,24,10,25,26);(1,3)(2,10)(4,17)(5,24)(6,19)(7,9)(8,20)(11,28)(12,21)(13,30)(14,31)(15,27)(16,26)(18,25)(22,23)(29,32);(1,4)(2,10)(3,17)(5,19)(6,24)(7,23)(8,18)(9,22)(11,16)(12,31)(13,29)(14,21)(15,27)(20,25)(26,28)(30,32)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,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231603, #72231704, #72231714, … |
| maximal normal subgroup of | #72231603, #72231704, #72231714, … |
| maximal quotient of | C2×#130780, #72250840, #72280163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37976);