Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60741 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,22,26,23,20,21,25)(19,27,29,28,31,32,30,24);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,19,23,31)(18,24,20,28)(21,29,22,30)(25,27,26,32);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,23)(19,28)(21,26)(22,25)(24,31)(27,30)(29,32);(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,28,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106920);