Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,24,38,23,29,22,28)(19,25,32,41,34,43,33,42)(20,35,36,26,21,27,37,44)(30,45,47,39,31,40,46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,30)(21,31)(22,33)(23,34)(24,32)(26,39)(27,40)(28,42)(29,43)(35,45)(36,47)(37,46)(38,41)(44,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,29,44)(19,30,34,31)(22,36,24,37)(25,39,43,48)(27,38,35,28)(32,46,33,47)(40,41,45,42);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,27,29,35)(19,31,34,30)(22,36,24,37)(25,40,43,45)(26,38,44,28)(32,46,33,47)(39,41,48,42);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,29,38)(19,32,34,33)(20,36,21,37)(25,41,43,42)(26,27,44,35)(30,46,31,47)(39,45,48,40)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6922250);