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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1544 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)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,33,41)(20,25,35,43)(21,26,22,27)(29,37,46,48)(30,38,45,47)(31,39,32,40)(34,42,36,44);(2,6)(7,8)(11,16)(13,15)(17,19)(18,24)(20,29)(21,32)(22,31)(23,33)(25,37)(26,40)(27,39)(28,41)(30,34)(35,46)(36,45)(38,42)(43,48)(44,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,37)(26,44)(27,42)(28,43)(30,31)(32,45)(33,46)(38,39)(40,47)(41,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,38)(26,42)(27,44)(28,43)(29,31)(32,46)(33,45)(37,39)(40,48)(41,47);(1,4,5,3)(2,7,6,8)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,35,36)(24,39,41,40)(25,42,43,44)(29,45,46,30)(37,47,48,38)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, 7442970);