Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42778 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,26)(19,23,29,35)(20,24,31,37)(21,25,32,38)(27,33,40,44)(28,34,41,45)(30,36,42,46)(39,43,47,48);(1,2,5,6)(3,7,4,8)(19,27)(20,30)(21,32)(23,33)(24,36)(25,38)(28,41)(29,40)(31,42)(34,45)(35,44)(37,46);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,19,21,27,22,29,32,40)(18,23,25,33,26,35,38,44)(20,28,30,39,31,41,42,47)(24,34,36,43,37,45,46,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(20,31)(24,37)(28,41)(30,42)(34,45)(36,46)(39,47)(43,48);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(17,20,21,30,22,31,32,42)(18,24,25,36,26,37,38,46)(19,28,27,39,29,41,40,47)(23,34,33,43,35,45,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=g,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751522);