Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #36967 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1160 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,13,7)(4,9,5,8)(6,10)(12,16)(14,15)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,16)(9,15)(10,11)(20,28)(23,30)(26,32)(29,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283474, #72283821, #72284006, … |
| maximal normal subgroup of | #72283474, #72283821, #72284006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121079);