Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #38948 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,27)(19,28,31,38)(20,24,32,40)(21,25,35,26)(23,36,39,30)(29,43,44,48)(33,41,34,42)(37,45,47,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,32)(24,40)(29,44)(33,34)(37,47)(41,42)(43,48)(45,46);(2,6)(5,7)(10,13)(14,16)(17,19,21,30,22,31,35,36)(18,23,26,28,27,39,25,38)(20,29,34,46,32,44,33,45)(24,37,41,43,40,47,42,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,24,27,40)(19,29,31,44)(21,34,35,33)(23,37,39,47)(25,42,26,41)(28,43,38,48)(30,46,36,45);(3,8)(5,7)(10,13)(14,16)(17,21,22,35)(18,25,27,26)(19,30,31,36)(20,33,32,34)(23,38,39,28)(24,41,40,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=hi,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6691891);