Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122142 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6,10,13)(4,7,11,15)(5,8)(9,14)(12,16)(17,18)(19,27,30,23)(20,31)(21,34)(22,25)(24,33)(26,37)(28,44,45,48)(29,39,35,41)(32,42)(36,40)(38,46,43,47);(1,3)(2,6,8,13)(4,9)(5,10)(7,14,15,12)(11,16)(17,19)(18,23)(20,28)(21,35)(22,30)(24,38)(25,27)(26,41)(29,37)(31,44)(32,45)(33,47)(34,39)(36,46)(40,43)(42,48);(1,4)(2,7,8,15)(3,9)(5,11)(6,12,13,14)(10,16)(17,20)(18,24)(19,28)(21,36)(22,32)(23,38)(25,40)(26,42)(27,43)(29,47)(30,45)(31,34)(33,37)(35,46)(39,44)(41,48)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29338);