Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| 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 | #37445 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,33)(21,26)(22,37)(23,28)(25,35)(27,30)(29,45)(31,40)(32,43)(34,39)(36,44)(38,42)(41,46)(47,48);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,31)(18,24,28,40)(20,29,36,46)(21,22,32,30)(25,39,42,48)(26,37,43,27)(33,45,44,41)(34,38,47,35);(11,16)(13,15)(17,20,23,36)(18,25,28,42)(19,29,31,46)(21,34,32,47)(22,38,30,35)(24,39,40,48)(26,41,43,45)(27,44,37,33);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(17,21,23,32)(18,26,28,43)(19,30,31,22)(20,34,36,47)(24,27,40,37)(25,41,42,45)(29,35,46,38)(33,39,44,48);(2,5)(6,7)(19,31)(21,32)(24,40)(26,43)(29,46)(34,47)(39,48)(41,45)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6551799);