Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(3,4)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,37)(28,34)(29,43)(30,35)(31,46)(32,40)(33,42)(36,45)(38,52)(39,48)(41,49)(44,51)(47,55)(50,54)(53,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,30,40)(26,32,35,37)(28,39,29,38)(31,41,45,56)(33,50,51,47)(34,52,43,48)(36,49,46,53)(42,55,44,54);(2,6)(7,8)(25,28,30,29)(26,33,35,51)(27,38,40,39)(31,44,45,42)(32,47,37,50)(34,36,43,46)(41,55,56,54)(48,49,52,53);(1,3,5,4)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,29,30,28)(26,34,35,43)(27,39,40,38)(31,44,45,42)(32,48,37,52)(33,36,51,46)(41,55,56,54)(47,49,50,53);(1,4,5,3)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,35)(28,29)(32,37)(36,46)(38,39)(42,44)(49,53)(54,55)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,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, 7134818);