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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(17,18,22,26)(19,27,28,44)(20,24,34,43)(21,35,36,25)(23,37,38,30)(29,45,31,48)(32,42,33,41)(39,46,40,47);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(19,28)(23,38)(27,44)(29,31)(30,37)(39,40)(45,48)(46,47);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,28,36,37)(18,23,25,27,26,38,35,44)(20,31,33,46,34,29,32,47)(24,40,42,48,43,39,41,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,22,34)(18,24,26,43)(19,29,28,31)(21,33,36,32)(23,39,38,40)(25,42,35,41)(27,45,44,48)(30,47,37,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,36)(18,25,26,35)(19,30,28,37)(20,32,34,33)(23,27,38,44)(24,41,43,42)(29,46,31,47)(39,48,40,45)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fg,d^2=fi,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=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6940030);