Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187195 |
| 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 | #62755 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,10,7,11,5,19)(3,12,14,23,15,8,20,24)(4,16,17,25,18,9,21,26)(13,27,30,32,29,22,28,31)(33,34)(35,44)(36,48,51,41)(37,42)(38,52)(39,43)(40,46)(45,59,61,58)(47,55)(49,63,50,57)(53,56)(54,60,62,64);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,22)(10,23)(11,12)(16,27)(17,28)(18,29)(19,24)(21,30)(25,31)(26,32)(33,35)(34,40)(36,45)(37,46)(38,53)(39,47)(41,54)(42,55)(43,56)(44,52)(48,62)(49,60)(50,64)(51,61)(57,59)(58,63);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,16)(12,27)(14,28)(15,29)(19,26)(20,30)(23,31)(24,32)(33,36)(34,41)(35,45)(37,49)(38,50)(39,51)(40,54)(42,57)(43,48)(44,58)(46,60)(47,61)(52,63)(53,64)(55,59)(56,62);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,18,21)(8,23,12,24)(9,25,16,26)(13,28,29,30)(22,31,27,32)(33,37,39,38)(34,42,43,52)(35,46,47,53)(36,49,51,50)(40,55,56,44)(41,57,48,63)(45,60,61,64)(54,59,62,58)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94570);