Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,8)(9,10,13,17)(11,15,18,14)(12,20,22,16)(19,23,21,24)(25,26,31,36)(27,37,38,48)(28,33,44,51)(29,35,30,34)(32,40,46,41)(39,53,42,56)(43,52,45,50)(47,55,49,54);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23)(27,38)(32,46)(37,48)(39,42)(40,41)(47,49)(53,56)(54,55);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,27,30,41,31,38,29,40)(26,32,35,37,36,46,34,48)(28,42,43,55,44,39,45,54)(33,49,50,56,51,47,52,53);(2,5)(6,8)(11,18)(14,15)(19,21)(23,24)(25,28)(26,33)(27,39)(29,45)(30,43)(31,44)(32,47)(34,52)(35,50)(36,51)(37,53)(38,42)(40,55)(41,54)(46,49)(48,56);(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)(25,29,31,30)(26,34,36,35)(27,40,38,41)(28,43,44,45)(32,48,46,37)(33,50,51,52)(39,54,42,55)(47,53,49,56)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fghi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105313);