Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(17,18,22,27)(19,28,31,43)(20,34,38,44)(21,25,29,26)(23,32,40,33)(24,36,42,37)(30,41,35,39)(45,47,48,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,29,32,22,31,21,33)(18,23,25,43,27,40,26,28)(20,35,36,46,38,30,37,47)(24,41,44,45,42,39,34,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,29,36,22,38,21,37)(18,24,25,44,27,42,26,34)(19,30,32,47,31,35,33,46)(23,39,43,48,40,41,28,45);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,31)(23,40)(28,43)(30,35)(32,33)(39,41)(45,48)(46,47);(11,15)(14,16)(17,21,22,29)(18,25,27,26)(19,32,31,33)(20,36,38,37)(23,28,40,43)(24,34,42,44)(30,46,35,47)(39,48,41,45)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6464754);