Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,20)(3,12,16,26,15,8,14,23)(4,17,18,24,19,9,21,25)(13,27,28,31,29,22,30,32)(33,34)(35,43,46,57)(36,40)(37,42)(38,41)(39,45,53,49)(44,60,62,59)(47,55)(48,54)(50,58)(51,56)(52,61,64,63);(1,3)(2,8)(4,13)(5,16)(6,14)(7,15)(9,22)(10,26)(11,12)(17,27)(18,30)(19,29)(20,23)(21,28)(24,32)(25,31)(33,35)(34,39)(36,44)(37,49)(38,46)(40,52)(41,53)(42,57)(43,56)(45,51)(47,63)(48,62)(50,61)(54,64)(55,59)(58,60);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,27)(14,28)(15,29)(16,30)(20,25)(23,31)(26,32)(33,36,38,48)(34,40,41,54)(35,44,46,62)(37,50,51,47)(39,52,53,64)(42,58,56,55)(43,59,57,60)(45,63,49,61);(1,5,7,6)(2,10,11,20)(3,14,15,16)(4,18,19,21)(8,23,12,26)(9,24,17,25)(13,28,29,30)(22,31,27,32)(34,41)(39,53)(40,54)(42,56)(43,57)(52,64)(55,58)(59,60)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94485);