Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349901 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,26)(4,9,18,24)(5,11,6,10)(8,20,22,15)(14,27,30,31)(17,25,19,23)(21,28,32,29)(33,34,39,42)(35,45,49,56)(36,50,51,58)(37,44,38,43)(40,48,55,47)(41,52,57,53)(46,60,61,54)(59,62,64,63);(1,3,7,16)(2,8,12,22)(4,14,18,30)(5,15,6,20)(9,21,24,32)(10,13,11,26)(17,28,19,29)(23,31,25,27)(33,35,39,49)(34,40,42,55)(36,46,51,61)(37,48,38,47)(41,54,57,60)(43,56,44,45)(50,64,58,59)(52,63,53,62);(1,4)(2,9)(3,14)(5,19)(6,17)(7,18)(8,21)(10,25)(11,23)(12,24)(13,27)(15,29)(16,30)(20,28)(22,32)(26,31)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,57)(43,50)(44,58)(45,59)(47,63)(48,62)(49,61)(55,60)(56,64);(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,17,18,19)(8,13,22,26)(9,23,24,25)(14,28,30,29)(21,31,32,27)(34,42)(36,51)(40,55)(43,44)(45,56)(46,61)(52,53)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257152);