Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347544 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62826 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,37)(4,18,22,9)(5,23,25,53)(6,10,20,32)(8,27,29,16)(11,34,36,26)(14,39,43,38)(15,35,41,60)(19,45,48,31)(21,49,52,54)(24,30,55,58)(28,42,44,56)(33,47,46,51)(40,61,63,59)(50,62,64,57);(1,3)(2,8)(4,14)(5,24)(6,15)(7,17)(9,28)(10,30)(11,35)(12,29)(13,23)(16,26)(18,44)(19,40)(20,41)(21,50)(22,43)(25,55)(27,34)(31,57)(32,58)(33,59)(36,60)(37,53)(38,54)(39,49)(42,47)(45,62)(46,61)(48,63)(51,56)(52,64);(4,19)(5,25)(9,31)(11,36)(13,37)(14,40)(16,27)(18,45)(21,51)(22,48)(23,53)(24,55)(26,34)(28,57)(33,49)(35,60)(38,61)(39,59)(42,64)(43,63)(44,62)(46,54)(47,52)(50,56);(1,4,20,48)(2,9,32,45)(3,14,41,63)(5,21,34,51)(6,19,7,22)(8,28,58,62)(10,31,12,18)(11,33,53,49)(13,38,60,61)(15,40,17,43)(16,42,55,64)(23,54,36,46)(24,50,27,56)(25,52,26,47)(29,44,30,57)(35,59,37,39);(2,10)(3,15)(9,31)(11,23)(12,32)(14,40)(16,24)(17,41)(18,45)(27,55)(33,54)(36,53)(42,50)(43,63)(46,49)(56,64)Polycyclic presentation:
2^9|a^2=i,d^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254762);