Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130294 |
| 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 | #62334 | 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 | |
620 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 620 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:
(1,2,7,14,6,26,3,16,9,15,18,30,8,13,19,32)(4,20,22,31,17,29,5,24,23,10,27,28,21,11,25,12);(1,3)(2,10)(4,5)(6,7)(8,18)(9,19)(11,13)(12,14)(15,20)(16,28)(17,22)(21,27)(23,25)(24,30)(26,29)(31,32);(1,4)(2,11)(3,5)(6,17)(7,22)(8,21)(9,23)(10,13)(12,14)(15,29)(16,31)(18,27)(19,25)(20,26)(24,30)(28,32)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,e^d=ei,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231615, #72231620, … |
| maximal normal subgroup of | #72231596, #72231615, #72231620, … |
| maximal quotient of | #72250564, #72250631, C2×#130296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37492);