Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343524 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62838 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,12,7,11,5,10)(3,13,15,26,16,8,20,22)(4,9,17,23,18,24,19,25)(14,27,29,31,30,21,28,32)(33,34)(35,44)(36,41)(37,42)(38,51)(39,43)(40,46)(45,59)(47,54)(48,56)(49,57)(50,58)(52,55)(53,62)(60,63)(61,64);(1,3,7,16)(2,8,11,13)(4,14,18,30)(5,15,6,20)(9,21,24,27)(10,22,12,26)(17,28,19,29)(23,31,25,32)(33,35,39,47)(34,40,43,55)(36,45,49,61)(37,46,38,52)(41,53,57,63)(42,54,51,44)(48,60,50,62)(56,59,58,64);(1,4)(2,9)(3,14)(5,19)(6,17)(7,18)(8,21)(10,25)(11,24)(12,23)(13,27)(15,29)(16,30)(20,28)(22,32)(26,31)(33,36)(34,41)(35,45)(37,50)(38,48)(39,49)(40,53)(42,58)(43,57)(44,59)(46,62)(47,61)(51,56)(52,60)(54,64)(55,63);(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,48,49,50)(40,54,55,44)(41,56,57,58)(45,60,61,62)(53,59,63,64);(1,5,7,6)(2,10,11,12)(3,15,16,20)(4,17,18,19)(8,22,13,26)(9,23,24,25)(14,28,30,29)(21,31,27,32)(36,49)(41,57)(45,61)(48,50)(53,63)(56,58)(59,64)(60,62)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=i,g^2=hi,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251403);