Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7504676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 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)(3,15,19,9)(4,20,23,40)(5,24,27,41)(6,13,7,12)(10,29,36,22)(11,30,38,26)(16,42,45,50)(17,47,25,56)(18,33,28,39)(21,35,55,60)(31,46,34,44)(32,48,37,49)(43,57,52,58)(51,53,59,54)(61,62,64,63);(1,3,7,28,8,19,6,18)(2,9,13,39,14,15,12,33)(4,16,22,44,23,45,29,46)(5,25,26,49,27,17,30,48)(10,34,20,50,36,31,40,42)(11,37,24,47,38,32,41,56)(21,52,54,62,55,43,53,63)(35,57,59,64,60,58,51,61);(1,4)(2,10)(3,16,19,45)(5,21)(6,22)(7,29)(8,23)(9,31,15,34)(11,35)(12,20)(13,40)(14,36)(17,43,25,52)(18,44,28,46)(24,51)(26,53)(27,55)(30,54)(32,57,37,58)(33,42,39,50)(38,60)(41,59)(47,64,56,61)(48,62,49,63);(1,5)(2,11)(3,17)(4,21)(6,26)(7,30)(8,27)(9,32)(10,35)(12,24)(13,41)(14,38)(15,37)(16,43)(18,48)(19,25)(20,51)(22,53)(23,55)(28,49)(29,54)(31,57)(33,56)(34,58)(36,60)(39,47)(40,59)(42,61)(44,62)(45,52)(46,63)(50,64);(1,6,8,7)(2,12,14,13)(3,18,19,28)(4,22,23,29)(5,26,27,30)(9,33,15,39)(10,20,36,40)(11,24,38,41)(16,44,45,46)(17,48,25,49)(21,53,55,54)(31,42,34,50)(32,56,37,47)(35,51,60,59)(43,62,52,63)(57,61,58,64)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,h^a=hi,c^b=ch,d^b=df,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411875);