Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,23)(22,35)(24,30)(26,34)(28,42)(29,41)(31,37)(32,46)(33,48)(36,45)(38,44)(39,47)(40,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(24,36)(26,35)(27,41)(28,32)(33,43)(44,47);(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,29,34,45)(18,24,26,27,23,36,35,41)(20,32,33,47,31,28,43,44)(25,39,40,46,37,38,48,42);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,21,31)(18,25,23,37)(19,28,29,32)(22,33,34,43)(24,38,36,39)(26,40,35,48)(27,42,41,46)(30,44,45,47);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(20,31)(25,37)(28,32)(33,43)(38,39)(40,48)(42,46)(44,47)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 6937084);