Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1435 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,26)(21,39)(22,40)(23,29)(24,27)(25,33)(28,32)(31,46)(34,44)(35,48)(36,42)(37,45)(38,43)(41,47);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,23,33)(18,25,29,30)(20,34,37,36)(21,32,24,22)(26,42,45,44)(27,28,39,40)(31,35,47,38)(41,48,46,43);(10,14)(13,16)(17,20,24,38,23,37,21,35)(18,26,27,43,29,45,39,48)(19,31,32,34,33,47,22,36)(25,41,40,42,30,46,28,44);(2,6)(5,7)(17,21,23,24)(18,27,29,39)(19,32,33,22)(20,35,37,38)(25,28,30,40)(26,43,45,48)(31,34,47,36)(41,44,46,42);(3,8)(5,7)(17,22,21,33,23,32,24,19)(18,28,39,30,29,40,27,25)(20,36,35,47,37,34,38,31)(26,44,48,46,45,42,43,41)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719478);