Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1363 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)(3,7)(4,6)(5,8)(17,18,23,28)(19,26,32,42)(20,25,33,39)(21,36,31,24)(22,37,29,27)(30,40,44,48)(34,46,45,38)(35,47,43,41);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,29,21,23,32,22,31)(18,24,37,42,28,36,27,26)(20,30,43,34,33,44,35,45)(25,38,47,48,39,46,41,40);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,33)(24,38)(26,40)(27,41)(28,39)(29,43)(31,45)(32,44)(36,46)(37,47)(42,48);(2,6)(5,7)(10,14)(13,16)(20,33)(25,39)(30,44)(34,45)(35,43)(38,46)(40,48)(41,47);(2,6)(5,7)(17,21,29,32,23,31,22,19)(18,26,37,24,28,42,27,36)(20,34,43,44,33,45,35,30)(25,40,47,38,39,48,41,46)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7507606);