Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,35,30,36,29,46)(27,37,42,54,41,32,40,50)(28,43,44,52,45,34,47,33)(38,53,55,48,39,49,56,51);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,32)(28,39)(29,40)(30,41)(31,42)(33,51)(34,49)(35,50)(36,37)(38,45)(43,53)(44,56)(46,54)(47,55)(48,52);(1,2)(3,8)(4,7)(5,6)(25,28,30,45)(26,33,36,52)(27,38,41,39)(29,44,31,47)(32,48,37,51)(34,46,43,35)(40,55,42,56)(49,50,53,54);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,39)(29,47)(30,45)(31,44)(32,49)(33,35)(36,43)(37,53)(38,41)(40,55)(42,56)(46,52)(48,54)(50,51);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,36,46)(27,40,41,42)(28,44,45,47)(32,50,37,54)(33,34,52,43)(38,55,39,56)(48,53,51,49)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397823);