Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | |
1384 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)(25,26,31,44,30,33,29,35)(27,36,43,32,37,51,39,47)(28,34,45,56,41,48,42,50)(38,52,55,46,40,54,53,49);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,40)(29,43)(30,37)(31,39)(33,47)(34,49)(35,51)(36,44)(38,41)(42,53)(45,55)(46,48)(50,52)(54,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,33)(27,37)(28,41)(35,44)(39,43)(42,45)(46,49)(52,54);(11,18)(14,22)(17,19)(21,23)(25,28,30,41)(26,34,33,48)(27,38,37,40)(29,42,31,45)(32,46,47,49)(35,50,44,56)(36,52,51,54)(39,53,43,55);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(26,33)(28,41)(32,47)(35,44)(36,51)(38,40)(42,45)(53,55)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388311);