Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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)(17,18,22,27)(19,28,23,24)(20,31,34,38)(21,35,36,26)(25,29,41,32)(30,39,37,42)(33,46,47,44)(40,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,44,48,46);(10,14)(11,15)(17,20,19,29,22,34,23,32)(18,25,24,38,27,41,28,31)(21,33,30,43,36,47,37,45)(26,40,39,46,35,48,42,44);(2,6)(5,7)(17,21)(18,26)(19,30)(20,33)(22,36)(23,37)(24,39)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(38,46)(41,48);(3,8)(5,7)(11,15)(13,16)(20,34)(21,36)(25,41)(26,35)(29,32)(30,37)(31,38)(39,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6668984);