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 | |
1371 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)(25,26,29,34,30,32,42,46)(27,35,37,43,38,48,41,31)(28,33,40,47,39,45,52,55)(36,49,51,54,50,56,53,44);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,38)(26,31,32,43)(28,36,39,50)(29,41,42,37)(33,44,45,54)(34,48,46,35)(40,53,52,51)(47,56,55,49);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(31,43)(33,45)(34,46)(35,48)(44,54)(47,55)(49,56);(11,18)(14,22)(17,19)(21,23)(25,28)(26,33)(27,36)(29,40)(30,39)(31,44)(32,45)(34,47)(35,49)(37,51)(38,50)(41,53)(42,52)(43,54)(46,55)(48,56);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(28,39)(33,45)(36,50)(40,52)(44,54)(47,55)(49,56)(51,53)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388353);