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 | #62743 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,10,6,9,20,24)(3,11,14,25,13,7,18,22)(4,15,17,26,16,8,19,23)(12,27,29,32,28,21,30,31)(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)(2,7)(4,12)(5,18)(6,13)(8,21)(9,11)(10,25)(14,20)(15,27)(16,28)(17,30)(19,29)(22,24)(23,32)(26,31)(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)(2,8)(3,12)(5,19)(6,16)(7,21)(9,15)(10,26)(11,27)(13,28)(14,30)(17,20)(18,29)(22,32)(23,24)(25,31)(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);(2,9)(7,11)(8,15)(10,24)(21,27)(22,25)(23,26)(31,32)(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=i,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, 94571);