Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,33)(27,29)(30,43)(31,37)(32,41)(34,40)(35,38)(36,42)(39,44)(45,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,23)(24,37)(25,38)(26,27)(28,41)(39,47)(40,42)(43,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19,29,31,22,32,21,33)(18,24,26,41,23,37,27,28)(20,30,34,46,35,45,36,44)(25,39,42,48,38,47,40,43);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20,22,35)(18,25,23,38)(19,30,32,45)(21,36,29,34)(24,39,37,47)(26,42,27,40)(28,43,41,48)(31,46,33,44);(17,21,22,29)(18,26,23,27)(19,31,32,33)(20,34,35,36)(24,28,37,41)(25,40,38,42)(30,44,45,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367845);