Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2202 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)(3,6)(4,5)(7,8)(17,18,24,30)(19,26,33,39)(20,25,32,40)(21,27,35,42)(22,29,23,28)(31,41,45,48)(34,38,46,47)(36,44,37,43);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,20,24,33,23,32)(18,25,28,26,30,40,29,39)(21,31,36,34,35,45,37,46)(27,38,43,41,42,47,44,48);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,24,32,23,19)(18,26,28,40,30,39,29,25)(21,34,36,45,35,46,37,31)(27,41,43,47,42,48,44,38);(11,16)(13,15)(17,21)(18,27)(19,31)(20,34)(22,36)(23,37)(24,35)(25,38)(26,41)(28,43)(29,44)(30,42)(32,46)(33,45)(39,48)(40,47);(10,14)(13,15)(21,35)(27,42)(31,45)(34,46)(36,37)(38,47)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413698);