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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,38)(28,34)(29,47)(30,48)(31,37)(32,35)(33,41)(36,40)(39,54)(42,52)(43,56)(44,50)(45,53)(46,51)(49,55);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,33)(28,42)(29,40)(30,32)(31,41)(34,50)(35,36)(37,38)(39,46)(43,55)(44,45)(47,48)(49,51)(52,53)(54,56);(2,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,29,43,31,45,32,46)(26,34,47,56,37,53,35,51)(27,39,30,44,41,55,40,42)(33,49,36,52,38,54,48,50);(10,16)(14,17)(15,22)(21,23)(25,29,31,32)(26,35,37,47)(27,40,41,30)(28,43,45,46)(33,36,38,48)(34,51,53,56)(39,42,55,44)(49,52,54,50);(11,19)(14,17)(18,24)(21,23)(25,30,32,27,31,40,29,41)(26,36,35,33,37,48,47,38)(28,44,46,39,45,42,43,55)(34,52,51,49,53,50,56,54)Polycyclic presentation:
2^9|b^2=g,c^2=fh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6719741);