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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,28,31,24)(20,32,36,25)(21,37,23,26)(29,40,34,44)(30,41,38,42)(33,43,47,48)(35,45,39,46);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,28)(20,33,36,47)(21,30,23,38)(25,43,32,48)(26,41,37,42)(29,39,34,35)(40,45,44,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,34)(21,38)(22,31)(23,30)(25,44)(26,42)(27,28)(29,36)(32,40)(33,39)(35,47)(37,41)(43,45)(46,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,29)(21,35)(22,36)(23,39)(24,40)(26,45)(27,32)(28,44)(30,47)(31,34)(33,38)(37,46)(41,43)(42,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,37)(19,30,31,38)(20,35,36,39)(24,41,28,42)(25,45,32,46)(29,47,34,33)(40,43,44,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7496889);