Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213744 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1956 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,33)(21,31)(23,29)(24,30)(25,28)(26,32)(27,35)(34,39)(36,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15,18,19)(12,20)(13,25,26,21)(14,22)(16,27,30,33)(17,31)(23,36,37,34)(24,35)(28,32)(29,39)(38,40);(2,6)(5,8)(9,12,14,24)(10,16,18,30)(11,20,22,35)(13,23,26,37)(15,27,19,33)(17,29,32,40)(21,36,25,34)(28,39,31,38);(3,7)(5,8)(9,13)(10,17)(11,21)(12,23)(14,26)(15,28)(16,29)(18,32)(19,31)(20,34)(22,25)(24,37)(27,38)(30,40)(33,39)(35,36)Polycyclic presentation:
2^9|b^2=h,c^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210036, #72210037, #72227719, … |
| maximal normal subgroup of | #72210036, #72210037, #72227719, … |
| maximal quotient of | #72240912, #72240915, #72240921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298906);