Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1713 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)(17,18,24,30)(19,25,20,26)(21,27,37,45)(22,28,23,29)(31,39,33,41)(32,40,34,42)(35,43,47,48)(36,44,38,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,24)(21,35)(22,32)(23,34)(26,30)(27,43)(28,40)(29,42)(31,36)(33,38)(37,47)(39,44)(41,46)(45,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,24,19)(18,26,30,25)(21,33,37,31)(22,32,23,34)(27,41,45,39)(28,40,29,42)(35,36,47,38)(43,44,48,46);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,21)(18,27)(19,31)(20,33)(22,36)(23,38)(24,37)(25,39)(26,41)(28,44)(29,46)(30,45)(32,47)(34,35)(40,48)(42,43);(1,4,5,3)(2,7,8,6)(17,22,24,23)(18,28,30,29)(19,32,20,34)(21,36,37,38)(25,40,26,42)(27,44,45,46)(31,47,33,35)(39,48,41,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7438716);