Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
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)(25,26,30,35)(27,31,38,47)(28,33,40,51)(29,42,44,34)(32,43,49,41)(36,50,37,46)(39,55,54,48)(45,53,56,52);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,38)(26,31,35,47)(28,36,40,37)(29,39,44,54)(32,45,49,56)(33,50,51,46)(34,48,42,55)(41,52,43,53);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,32)(27,36)(29,43)(30,40)(31,45)(33,42)(34,51)(35,49)(37,38)(39,53)(41,44)(46,48)(47,56)(50,55)(52,54);(2,6)(3,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,33)(27,37)(29,41)(30,40)(31,46)(32,42)(34,49)(35,51)(36,38)(39,53)(43,44)(45,48)(47,50)(52,54)(55,56);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(27,38)(28,40)(31,47)(32,49)(33,51)(39,54)(41,43)(48,55)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,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, 7209384);