Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2416 |
| 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 | |
2433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,30,8,13,6,12)(3,14,29,9,18,33,17,34)(4,19,24,10,22,38,23,37)(5,11,27,41,28,42,26,40)(15,31,20,35,46,50,47,43)(16,44,25,32,49,39,48,57)(21,51,55,36,53,60,54,59)(45,56,52,58,62,64,63,61);(1,3)(2,9)(4,20)(5,25)(6,29)(7,17)(8,18)(10,35)(11,39)(12,33)(13,34)(14,30)(15,24)(16,27)(19,50)(21,45)(22,47)(23,46)(26,49)(28,48)(31,38)(32,41)(36,56)(37,43)(40,57)(42,44)(51,58)(52,54)(53,62)(55,63)(59,64)(60,61);(1,4,7,24,8,22,6,23)(2,10,12,19,13,37,30,38)(3,15,17,47,18,46,29,20)(5,21,27,55,28,53,26,54)(9,31,33,35,34,50,14,43)(11,36,40,51,42,59,41,60)(16,45,48,63,49,62,25,52)(32,56,39,58,57,64,44,61);(1,5)(2,11)(3,16)(4,21)(6,26)(7,27)(8,28)(9,32)(10,36)(12,40)(13,42)(14,44)(15,45)(17,48)(18,49)(19,51)(20,52)(22,53)(23,54)(24,55)(25,29)(30,41)(31,56)(33,39)(34,57)(35,58)(37,59)(38,60)(43,61)(46,62)(47,63)(50,64);(4,22)(10,37)(15,46)(19,38)(20,47)(21,53)(23,24)(31,50)(35,43)(36,59)(45,62)(51,60)(52,63)(54,55)(56,64)(58,61)Polycyclic presentation:
2^9|a^2=fi,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350258);