Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,34)(29,35)(30,44)(31,33)(32,39)(37,51)(38,45)(40,50)(41,48)(42,47)(43,49)(46,52)(53,56)(54,55);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,30,39,31,40,29,38)(26,32,35,36,33,45,44,50)(28,37,41,52,42,53,43,54)(34,46,49,51,47,55,48,56);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(26,33)(32,45)(34,47)(35,44)(36,50)(46,55)(48,49)(51,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,28)(26,34)(27,37)(29,43)(30,41)(31,42)(32,46)(33,47)(35,49)(36,51)(38,54)(39,52)(40,53)(44,48)(45,55)(50,56);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(25,29,31,30)(26,35,33,44)(27,38,40,39)(28,41,42,43)(32,36,45,50)(34,48,47,49)(37,52,53,54)(46,56,55,51)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394267);