Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#121 |
| prime radical | #19160 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,10)(4,7)(9,15)(12,13)(14,16)(17,18,21,25,22,19,26,24)(20,23,28,31,29,27,32,30);(1,2,7,15,5,9,4,6)(3,11,10,14,12,8,13,16);(1,3)(2,8)(4,13)(5,12)(6,14)(7,10)(9,11)(15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,31)(30,32);(3,12)(8,11)(10,13)(14,16)(17,20,21,28,22,29,26,32)(18,23,25,31,19,27,24,30)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287402, #72322056, #72322165 |
| maximal normal subgroup of | #72287402, #72322056, #72322165 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 119175);