Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,32,23)(19,24,30,37,29,38,42,34)(20,25,31,40,33,39,44,35)(28,36,41,46,43,47,48,45);(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(21,32)(24,34)(26,27)(28,41)(30,42)(33,44)(35,40)(37,38)(43,48)(45,46);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,22,29)(18,24,26,38)(20,28,33,43)(21,30,32,42)(23,34,27,37)(25,36,39,47)(31,41,44,48)(35,45,40,46);(11,16)(13,15)(17,20,32,44,22,33,21,31)(18,25,23,35,26,39,27,40)(19,28,42,48,29,43,30,41)(24,36,34,45,38,47,37,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,26)(19,29)(23,27)(25,39)(28,43)(30,42)(35,40)(41,48)Polycyclic presentation:
2^9|a^2=fg,c^2=ghi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6999585);