Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | |
1353 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)(25,26,29,35,30,34,44,51)(27,36,38,47,39,52,43,31)(28,33,42,50,41,32,48,49)(37,54,55,45,40,53,56,46);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,39)(26,31,34,47)(28,40,41,37)(29,43,44,38)(32,46,33,45)(35,52,51,36)(42,55,48,56)(49,53,50,54);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,32)(27,37)(29,42)(30,41)(31,45)(33,34)(35,49)(36,53)(38,55)(39,40)(43,56)(44,48)(46,47)(50,51)(52,54);(11,18)(14,22)(17,19)(21,23)(25,28)(26,33)(27,37)(29,42)(30,41)(31,46)(32,34)(35,50)(36,54)(38,55)(39,40)(43,56)(44,48)(45,47)(49,51)(52,53);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(26,34)(28,41)(31,47)(35,51)(36,52)(37,40)(42,48)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388361);