Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| 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 | C42 | C25 |
| prime radical | #62956 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,22)(4,9,17,25)(5,10,6,19)(8,20,23,14)(13,27,30,32)(16,24,18,26)(21,28,31,29)(33,34)(35,45)(36,42)(37,40)(38,44)(39,43)(41,48)(46,61)(47,57)(49,54)(50,64)(51,60)(52,55)(53,59)(56,63)(58,62);(1,3,7,15)(2,8,11,23)(4,13,17,30)(5,14,6,20)(9,21,25,31)(10,22,19,12)(16,28,18,29)(24,27,26,32)(34,40)(37,38)(41,54)(42,55)(43,44)(45,57)(47,48)(52,53)(56,64)(58,61)(59,60)(62,63);(1,4)(2,9)(3,13)(5,18)(6,16)(7,17)(8,21)(10,26)(11,25)(12,27)(14,29)(15,30)(19,24)(20,28)(22,32)(23,31)(33,35,39,49)(34,41,43,57)(36,50,51,46)(37,48,38,47)(40,54,44,45)(42,58,60,56)(52,62,53,63)(55,61,59,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,26)(13,28,30,29)(21,27,31,32)(33,36,39,51)(34,42,43,60)(35,46,49,50)(37,52,38,53)(40,55,44,59)(41,56,57,58)(45,61,54,64)(47,62,48,63);(2,11)(8,23)(9,25)(10,19)(12,22)(21,31)(24,26)(27,32)(34,43)(36,51)(40,44)(41,57)(45,54)(46,50)(52,53)(62,63)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258532);