Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1744 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,33)(21,26)(22,27)(23,37)(24,31)(25,36)(29,44)(30,43)(32,39)(34,40)(35,41)(38,45)(42,47)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,38)(26,39)(27,28)(33,48)(34,45)(35,46)(36,47)(37,43)(40,42)(41,44);(11,15)(13,16)(17,20,22,35)(18,25,27,40)(19,29,31,46)(21,36,23,34)(24,38,28,42)(26,41,37,33)(30,47,32,45)(39,44,43,48);(11,15)(13,16)(17,21,22,23)(18,26,27,37)(19,30,31,32)(20,34,35,36)(24,39,28,43)(25,33,40,41)(29,45,46,47)(38,48,42,44);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,31)(20,35)(24,28)(25,40)(30,32)(33,41)(34,36)(39,43)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6379399);