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 | C23 | D4×Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(5,14)(8,9)(10,15)(11,13)(12,16)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,2,6,10)(3,8,4,12)(5,15,14,7)(9,11,16,13)(19,27)(23,31)(26,32)(28,29);(1,3,6,4)(2,8,10,12)(5,11,14,13)(7,9,15,16)(17,19,20,28,22,27,21,29)(18,23,24,26,25,31,30,32);(1,4,6,3)(2,9,10,16)(5,11,14,13)(7,8,15,12)(19,27)(23,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=i,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 | #72287574, #72287597, #72287604, … |
| maximal normal subgroup of | #72287574, #72287597, #72287604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132957);