Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1782 |
| prime radical | #62814 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #117 ◁ #3428 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,12,24)(4,15,17,7)(5,9)(6,10)(8,14,21,26)(13,27)(16,30,31,20)(18,22)(19,28)(23,25)(29,32);(2,7)(8,19)(9,20)(10,15)(11,23)(21,29)(22,30)(24,27);(1,3)(2,8,10,21)(4,13)(5,14)(6,12)(7,19,15,29)(9,11,22,24)(16,28)(17,25)(18,26)(20,23,30,27)(31,32);(3,12)(8,21)(11,24)(13,25)(14,26)(19,29)(23,27)(28,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=di,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280637, #72280847, #72292696, … |
| maximal normal subgroup of | #72280637, #72280847, #72292696, … |
| maximal quotient of | #72280621, #72280868, #72281167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400669);