Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #1598 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,23)(4,16,18,9)(5,10,6,20)(8,21,24,14)(13,28,31,27)(17,26,19,25)(22,30,32,29)(33,34,39,38)(35,40,46,45)(36,41,49,48)(37,42,50,51)(43,55,57,52)(44,53,58,59)(47,54,60,61)(56,63,64,62);(1,3)(2,8)(4,13)(5,14)(6,21)(7,15)(9,22)(10,23)(11,24)(12,20)(16,32)(17,29)(18,31)(19,30)(25,27)(26,28)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,52)(42,53)(47,56)(48,55)(49,57)(50,58)(51,59)(54,62)(60,64)(61,63);(33,36,39,49)(34,41,38,48)(35,43,46,57)(37,47,50,60)(40,52,45,55)(42,54,51,61)(44,56,58,64)(53,62,59,63);(1,4,7,18)(2,9,11,16)(3,13,15,31)(5,19,6,17)(8,22,24,32)(10,26,20,25)(12,27,23,28)(14,30,21,29)(33,37,39,50)(34,42,38,51)(35,44,46,58)(36,47,49,60)(40,53,45,59)(41,54,48,61)(43,56,57,64)(52,62,55,63);(1,5,7,6)(2,10,11,20)(3,14,15,21)(4,17,18,19)(8,23,24,12)(9,25,16,26)(13,29,31,30)(22,27,32,28)(33,38,39,34)(35,45,46,40)(36,48,49,41)(37,42,50,51)(43,55,57,52)(44,53,58,59)(47,54,60,61)(56,62,64,63)Polycyclic presentation:
2^9|a^2=hi,c^2=i,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260476);