Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,3,4,8)(5,9,14,16,15,11,12,13)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,3)(2,7)(4,10)(5,13)(6,8)(9,14)(11,12)(15,16)(19,27)(23,31)(26,32)(28,29);(1,4,7,6)(2,8,3,10)(5,12,15,14)(9,13,11,16)(17,19,20,28,22,27,21,29)(18,23,24,26,25,31,30,32);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(19,27)(23,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=g,c^2=fh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72287402, #72287426, #72287433, … |
| maximal normal subgroup of | #72287402, #72287426, #72287433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132986);