Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | |
1289 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)(32,40)(37,48)(38,46)(39,53)(42,52)(43,56)(44,51)(45,49)(47,55)(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)(25,27,30,40)(26,32,35,36)(28,38,31,37)(29,43,44,54)(33,46,41,48)(34,50,51,56)(39,42,55,45)(47,52,53,49);(1,4,5,3)(2,6,7,8)(27,37)(28,31)(32,46)(33,41)(36,48)(38,40)(39,54)(42,45)(43,55)(47,56)(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)(27,38,40,37)(29,42,44,45)(32,46,36,48)(34,49,51,52)(39,43,55,54)(47,56,53,50);(11,19)(14,17)(18,24)(22,23)(25,29,30,44)(26,34,35,51)(27,39,40,55)(28,42,31,45)(32,47,36,53)(33,49,41,52)(37,54,38,43)(46,56,48,50)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6598095);