Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1379 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,31,35)(27,36,40,53)(28,33,43,51)(29,34,30,45)(32,38,47,39)(37,52,41,56)(42,49,44,50)(46,55,48,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,20)(10,15,17,18)(12,19,13,21)(16,23,22,24);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27,30,39,31,40,29,38)(26,32,34,36,35,47,45,53)(28,41,42,55,43,37,44,54)(33,48,50,56,51,46,49,52);(2,6)(3,8)(11,20)(15,18)(19,21)(23,24)(25,28)(26,33)(27,37)(29,44)(30,42)(31,43)(32,46)(34,50)(35,51)(36,52)(38,55)(39,54)(40,41)(45,49)(47,48)(53,56);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(25,29,31,30)(26,34,35,45)(27,38,40,39)(28,42,43,44)(32,36,47,53)(33,49,51,50)(37,54,41,55)(46,56,48,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7242177);