Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1255 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)(25,26,31,35)(27,37,41,54)(28,33,29,34)(30,45,38,36)(32,46,49,42)(39,52,40,53)(43,51,44,50)(47,56,48,55);(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,38,46,31,41,30,42)(26,32,45,54,35,49,36,37)(28,40,44,55,29,39,43,56)(33,48,51,52,34,47,50,53);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,19)(22,23)(25,28)(26,33)(27,39)(29,31)(30,43)(32,47)(34,35)(36,50)(37,52)(38,44)(40,41)(42,55)(45,51)(46,56)(48,49)(53,54);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,31,28)(26,34,35,33)(27,40,41,39)(30,44,38,43)(32,48,49,47)(36,51,45,50)(37,53,54,52)(42,56,46,55);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,35)(27,41)(28,29)(36,45)(42,46)(43,44)(47,48)(52,53)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,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^b=ei,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394240);