Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
951 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)(3,13)(4,17,20,26)(5,10)(6,11)(7,12)(8,15)(9,18,25,19)(14,28,31,22)(16,23)(21,24)(27,30,32,29)(33,34,38,43,39,42,37,50)(35,40,46,56,45,54,52,55)(36,47,48,41,49,58,51,57)(44,59,62,63,60,64,61,53);(1,3,5,15,7,16,6,21)(2,8,11,13,12,24,10,23)(4,14,18,29,20,31,19,30)(9,22,26,27,25,28,17,32)(33,35,38,52,39,45,37,46)(34,40,50,56,42,54,43,55)(36,44,48,61,49,60,51,62)(41,53,47,59,57,63,58,64);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,22)(10,17)(11,26)(12,25)(13,27)(15,29)(16,31)(21,30)(23,32)(24,28)(33,36)(34,41)(35,44)(37,51)(38,48)(39,49)(40,53)(42,57)(43,58)(45,60)(46,62)(47,50)(52,61)(54,63)(55,64)(56,59);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,17,25,26)(14,29,31,30)(22,32,28,27)(34,42)(40,54)(41,57)(43,50)(47,58)(53,63)(55,56)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108005);