Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 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)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,25)(21,34)(22,38)(23,29)(24,28)(27,37)(30,40)(31,41)(32,43)(33,42)(35,44)(36,45)(39,46)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,23)(21,35)(22,31)(24,33)(26,29)(27,44)(28,41)(30,36)(32,39)(34,48)(37,47)(38,42)(40,46)(43,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,23,19)(18,26,29,25)(21,32,37,30)(22,31,24,33)(27,43,34,40)(28,41,38,42)(35,36,47,39)(44,46,48,45);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,21)(18,27)(19,30)(20,32)(22,36)(23,37)(24,39)(25,40)(26,43)(28,45)(29,34)(31,47)(33,35)(38,46)(41,44)(42,48);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,28,29,38)(19,31,20,33)(21,36,37,39)(25,41,26,42)(27,45,34,46)(30,47,32,35)(40,44,43,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7438673);