Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #37659 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1855 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,29,39)(20,31,33,47)(21,26,22,37)(25,34,43,35)(28,44,32,46)(30,40,36,41)(38,48,42,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,27,39)(20,32,33,28)(21,36,22,30)(25,42,43,38)(26,41,37,40)(31,46,47,44)(34,45,35,48);(10,14)(11,15)(17,20,21,34,23,33,22,35)(18,25,37,47,27,43,26,31)(19,28,36,48,29,32,30,45)(24,38,40,46,39,42,41,44);(2,6)(5,7)(19,29)(20,33)(24,39)(25,43)(30,36)(31,47)(34,35)(40,41);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,30,29,36)(20,34,33,35)(24,40,39,41)(25,31,43,47)(28,45,32,48)(38,46,42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,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=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6669273);