Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45243 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 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)(17,18,22,27)(19,28,23,24)(20,31,33,37)(21,26,36,42)(25,29,40,34)(30,41,35,38)(32,44,46,47)(39,43,48,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,35,36,30)(25,37,40,31)(26,41,42,38)(32,45,46,43)(39,44,48,47);(9,11)(10,13)(12,16)(14,15)(17,20,19,29,22,33,23,34)(18,25,24,37,27,40,28,31)(21,32,30,43,36,46,35,45)(26,39,38,47,42,48,41,44);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,22,36)(18,26,27,42)(19,30,23,35)(20,32,33,46)(24,38,28,41)(25,39,40,48)(29,43,34,45)(31,44,37,47);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(20,33)(21,36)(25,40)(26,42)(29,34)(30,35)(31,37)(38,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,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=dh,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, 6936888);