Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2246 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,34)(29,45)(30,40)(31,41)(33,48)(35,44)(36,42)(37,46)(38,47)(39,43);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,30)(18,24,27,40)(20,33,34,36)(21,35,29,39)(23,38,37,31)(25,42,32,48)(26,44,45,43)(28,47,46,41);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20)(18,25)(19,29)(21,30)(22,34)(23,39)(24,26)(27,32)(28,44)(31,36)(33,38)(35,37)(40,45)(41,48)(42,47)(43,46);(17,21,22,29)(18,26,27,45)(19,20,30,34)(23,33,37,36)(24,32,40,25)(28,48,46,42)(31,39,38,35)(41,43,47,44);(18,27)(19,30)(20,34)(26,45)(28,46)(31,38)(35,39)(42,48)Polycyclic presentation:
2^9|b^2=fg,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348224);