Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,35)(27,37,40,32)(28,34,43,52)(29,36,31,33)(38,50,41,47)(39,48,44,46)(42,49,45,51)(53,55,54,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,27)(26,32)(28,41)(29,44)(30,40)(31,39)(33,46)(34,50)(35,37)(36,48)(38,43)(42,53)(45,54)(47,52)(49,56)(51,55);(1,3)(2,5)(4,8)(6,7)(26,33)(29,31)(32,46)(34,49)(35,36)(37,48)(39,44)(42,45)(47,55)(50,56)(51,52)(53,54);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,47)(33,49)(35,52)(36,51)(37,50)(39,54)(40,41)(44,53)(46,55)(48,56);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,30,31)(26,33,35,36)(27,39,40,44)(28,42,43,45)(32,48,37,46)(34,51,52,49)(38,53,41,54)(47,55,50,56)Polycyclic presentation:
2^9|a^2=f,b^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7213848);