Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33691 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21122 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,14,5,12,8,23)(3,13,6,21,16,9,17,11)(4,18,19,25,20,26,24,30)(10,22,27,31,29,32,28,15);(1,3,7,6,5,16,8,17)(2,9,14,11,12,13,23,21)(4,15,19,22,20,31,24,32)(10,25,27,26,29,30,28,18);(1,4)(2,10)(3,15)(5,20)(6,22)(7,19)(8,24)(9,25)(11,26)(12,29)(13,30)(14,27)(16,31)(17,32)(18,21)(23,28);(1,3,7,6,5,16,8,17)(2,11,23,9,12,21,14,13)(4,15,19,22,20,31,24,32)(10,26,28,25,29,18,27,30)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fh,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal quotient of | #72314250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402195);