Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350717 |
| 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 | #62959 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,14,24)(4,8,15,22)(5,17,19,9)(7,18,21,13)(12,25,26,32)(16,29,28,23)(20,30,31,27)(33,34,39,44)(35,45,49,54)(36,42,51,59)(37,40,38,43)(41,47,57,48)(46,61,50,64)(52,55,53,60)(56,62,58,63);(1,3,6,14)(2,7,10,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(9,24,17,11)(16,30,28,27)(23,32,29,25)(34,40)(37,38)(41,54)(42,55)(43,44)(45,57)(47,48)(52,53)(56,64)(58,61)(59,60)(62,63);(1,4,6,15)(2,8,10,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(9,23,17,29)(11,25,24,32)(13,27,18,30)(33,35,39,49)(34,41,44,57)(36,50,51,46)(37,48,38,47)(40,54,43,45)(42,58,59,56)(52,62,53,63)(55,61,60,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(33,36)(34,42)(35,46)(37,52)(38,53)(39,51)(40,55)(41,56)(43,60)(44,59)(45,61)(47,62)(48,63)(49,50)(54,64)(57,58);(36,51)(42,59)(46,50)(52,53)(55,60)(56,58)(61,64)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257968);