Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14428 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
948 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)(3,11,13,27)(4,15,17,24)(5,9)(6,10)(7,14,22,25)(8,16,23,18)(12,28)(19,20)(21,29)(26,30)(31,32);(1,2)(3,7,13,22)(4,8)(5,9)(6,10)(11,25,27,14)(12,28,29,21)(15,16)(17,23)(18,24)(19,20)(26,31,32,30);(1,3,5,14,6,13,19,25)(2,7,20,27,10,22,9,11)(4,12,18,31,17,29,16,30)(8,21,15,32,23,28,24,26);(1,4)(2,8)(3,12)(5,18)(6,17)(7,21)(9,24)(10,23)(11,26)(13,29)(14,31)(15,20)(16,19)(22,28)(25,30)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efg,f^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293611);