Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1296 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)(3,7)(4,6)(5,8)(17,18,22,24)(19,28,29,45)(20,33,34,26)(21,27,23,38)(25,31,39,32)(30,46,35,44)(36,43,37,42)(40,47,41,48);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(18,24)(19,29)(20,34)(27,38)(31,32)(36,37)(40,41)(44,46);(10,13)(14,15)(17,19,21,31,22,29,23,32)(18,25,38,45,24,39,27,28)(20,35,37,47,34,30,36,48)(26,41,42,44,33,40,43,46);(2,6)(5,7)(17,20,22,34)(18,26,24,33)(19,30,29,35)(21,37,23,36)(25,40,39,41)(27,43,38,42)(28,44,45,46)(31,48,32,47);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,27,24,38)(19,31,29,32)(20,36,34,37)(25,28,39,45)(26,42,33,43)(30,47,35,48)(40,46,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gi,d^2=i,e^2=h,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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6711497);