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 | #23962 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1627 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)(17,18,22,23)(19,28,29,40)(20,32,33,25)(21,26,36,27)(24,38,37,31)(30,45,44,43)(34,42,35,41)(39,47,48,46);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,29)(20,33)(26,27)(31,38)(34,35)(39,48)(43,45);(2,6)(3,7)(10,13)(14,16)(17,19,21,31,22,29,36,38)(18,24,27,28,23,37,26,40)(20,30,35,47,33,44,34,46)(25,39,41,43,32,48,42,45);(9,11,12,15)(10,14,13,16)(17,20,22,33)(18,25,23,32)(19,30,29,44)(21,35,36,34)(24,39,37,48)(26,42,27,41)(28,43,40,45)(31,47,38,46);(3,7)(5,8)(17,21,22,36)(18,26,23,27)(19,31,29,38)(20,34,33,35)(24,40,37,28)(25,41,32,42)(30,46,44,47)(39,43,48,45)Polycyclic presentation:
2^9|a^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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670723);