Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2684 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,25,42)(5,11,29,41)(6,13,7,12)(9,19,34,18)(10,30,37,24)(16,35,22,31)(17,44,48,38)(23,53,57,51)(26,58,47,32)(27,40,28,39)(36,55,63,56)(43,49,50,46)(45,60,52,61)(54,64,59,62);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,24,49,25,16,30,46)(5,26,27,17,29,47,28,48)(10,35,21,50,37,31,42,43)(11,38,39,32,41,44,40,58)(23,54,56,45,57,59,55,52)(36,62,53,60,63,64,51,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31)(11,36)(12,21)(13,42)(14,37)(15,43)(17,45)(18,46)(19,49)(20,22)(26,59)(27,55)(28,56)(29,57)(32,60)(33,50)(34,35)(38,64)(39,51)(40,53)(41,63)(44,62)(47,54)(48,52)(58,61);(1,5)(2,11)(3,17)(4,23)(6,27)(7,28)(8,29)(9,32)(10,36)(12,39)(13,40)(14,41)(15,44)(16,45)(18,47)(19,26)(20,48)(21,51)(22,52)(24,55)(25,57)(30,56)(31,60)(33,38)(34,58)(35,61)(37,63)(42,53)(43,62)(46,54)(49,59)(50,64);(1,6,8,7)(2,12,14,13)(3,18,20,19)(4,24,25,30)(5,27,29,28)(9,15,34,33)(10,21,37,42)(11,39,41,40)(16,46,22,49)(17,47,48,26)(23,55,57,56)(31,43,35,50)(32,44,58,38)(36,51,63,53)(45,54,52,59)(60,62,61,64)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411491);