Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 | #62865 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,22)(3,14,30,48)(4,7,13,8)(5,11,31,49)(6,21,32,39)(10,33,44,17)(12,28,27,29)(15,40,35,26)(16,41,56,64)(18,38,19,42)(20,23,36,25)(24,50,57,53)(34,58,63,46)(37,51,55,54)(43,61,59,52)(45,62,47,60);(1,3,8,18)(2,10,4,15)(5,16,25,47)(6,26,29,17)(7,19,9,30)(11,34,20,43)(12,38,21,14)(13,35,22,44)(23,45,31,56)(24,52,54,46)(27,42,39,48)(28,33,32,40)(36,59,49,63)(37,60,50,41)(51,58,57,61)(53,64,55,62);(1,4,8,2)(3,15,18,10)(5,11,25,20)(6,27,29,39)(7,22,9,13)(12,28,21,32)(14,40,38,33)(16,34,47,43)(17,48,26,42)(19,44,30,35)(23,36,31,49)(24,53,54,55)(37,57,50,51)(41,58,60,61)(45,59,56,63)(46,62,52,64);(1,5)(2,11)(3,16)(4,20)(6,24)(7,23)(8,25)(9,31)(10,34)(12,37)(13,36)(14,41)(15,43)(17,46)(18,47)(19,45)(21,50)(22,49)(26,52)(27,55)(28,51)(29,54)(30,56)(32,57)(33,58)(35,59)(38,60)(39,53)(40,61)(42,62)(44,63)(48,64);(5,23)(11,36)(16,45)(20,49)(24,51)(25,31)(34,59)(37,53)(41,62)(43,63)(46,61)(47,56)(50,55)(52,58)(54,57)(60,64)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259636);