Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187442 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62827 | C2 |
| LCS | G ▷ C42 ▷ 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)(3,11,14,7)(4,15,16,24)(5,10)(6,9)(8,17,22,18)(12,27)(13,26,31,21)(19,23)(20,29)(25,30)(28,32)(33,34,38,44,39,42,37,43)(35,45,49,40,48,55,47,54)(36,41,50,57,51,56,52,58)(46,60,61,64,62,59,63,53);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,32)(16,28)(17,29)(18,30)(19,31)(22,25)(23,26)(24,27)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,54)(43,55)(44,45)(50,63)(51,62)(52,61)(56,64)(57,59)(58,60);(1,4)(2,8)(3,12)(5,18)(6,16)(7,20)(9,22)(10,24)(11,25)(13,30)(14,28)(15,23)(17,19)(21,27)(26,32)(29,31)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,56)(43,58)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60);(2,9)(4,16)(7,11)(10,23)(12,28)(17,18)(21,26)(29,30)(34,42)(40,54)(41,56)(43,44)(45,55)(53,64)(57,58)(59,60)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94763);