Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | #37701 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,22,27)(19,23,29,39)(20,32,35,44)(21,25,34,26)(24,36,42,37)(28,45,33,48)(30,40,31,43)(38,47,41,46);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,27,39)(20,33,35,28)(21,31,34,30)(24,41,42,38)(25,43,26,40)(32,48,44,45)(36,46,37,47);(2,6)(3,7)(10,13)(14,16)(17,20,34,36,22,35,21,37)(18,24,25,44,27,42,26,32)(19,28,30,47,29,33,31,46)(23,38,43,48,39,41,40,45);(9,11,12,15)(10,14,13,16)(19,29)(20,35)(23,39)(24,42)(30,31)(32,44)(36,37)(40,43);(3,7)(5,8)(17,21,22,34)(18,25,27,26)(19,30,29,31)(20,36,35,37)(23,40,39,43)(24,32,42,44)(28,46,33,47)(38,48,41,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670755);