Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,8)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,36)(28,41)(29,34)(30,35)(31,33)(32,39)(37,47)(38,53)(40,46)(42,52)(43,56)(44,51)(45,49)(48,55)(50,54);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,40)(29,43)(30,39)(31,37)(33,46)(34,50)(35,36)(38,45)(41,47)(42,55)(44,54)(48,49)(51,56)(52,53);(1,4,5,3)(2,6,7,8)(27,37)(28,31)(32,46)(33,41)(36,47)(38,54)(39,40)(42,45)(43,55)(48,56)(49,52)(50,53);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,30,31)(26,33,35,41)(27,37,39,40)(29,42,44,45)(32,47,36,46)(34,49,51,52)(38,54,55,43)(48,50,53,56);(11,19)(14,17)(18,24)(22,23)(25,29,30,44)(26,34,35,51)(27,38,39,55)(28,42,31,45)(32,48,36,53)(33,49,41,52)(37,54,40,43)(46,56,47,50)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6598209);