Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130232 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8350 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #62331 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,6,13,7,15,26,32,8,14,17,27,5,12,25,30)(3,16,19,11,18,31,24,9,20,28,4,21,23,10,22,29);(2,9)(4,22)(5,7)(6,26)(10,27)(11,15)(12,21)(13,31)(14,29)(16,32)(17,25)(18,23)(19,24)(28,30);(1,3,8,20)(2,10,14,31)(4,17,19,6)(5,23,7,18)(9,27,29,13)(11,32,21,30)(12,28,15,16)(22,25,24,26)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72231624, #72231708, #72231714, … |
| maximal normal subgroup of | #72231624, #72231708, #72231714, … |
| maximal quotient of | #72240800, #72250501, C2×#130233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37429);