Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 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)(9,10)(11,13)(12,14)(15,16)(17,18,24,30)(19,31,35,43)(20,36,38,42)(21,27,41,48)(22,28,23,29)(25,34,44,33)(26,32,45,37)(39,46,40,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,34,24,35,22,33)(18,25,28,43,30,44,29,31)(20,21,37,39,38,41,32,40)(26,27,36,47,45,48,42,46);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,32,24,38,23,37)(18,26,29,42,30,45,28,36)(19,21,33,40,35,41,34,39)(25,27,31,46,44,48,43,47);(11,15)(13,16)(17,21)(18,27)(19,32)(20,34)(22,40)(23,39)(24,41)(25,42)(26,43)(28,47)(29,46)(30,48)(31,45)(33,38)(35,37)(36,44);(10,14)(13,16)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,38,32)(21,39,41,40)(25,43,44,31)(26,42,45,36)(27,46,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380113);