Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350909 |
| 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 | #3460 | C22 |
| 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:
(1,2,7,12)(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)(33,34,39,44)(35,45,49,54)(36,50,51,42)(37,43,38,41)(40,48,56,47)(46,60,61,59)(52,57,53,58)(55,62,64,63);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,30)(19,29)(23,31)(24,32)(25,27)(33,35,39,49)(34,40,44,56)(36,46,51,61)(37,48,38,47)(41,54,43,45)(42,55,50,64)(52,63,53,62)(57,60,58,59);(1,4,7,18)(2,9,12,24)(3,14,16,30)(5,19,6,17)(8,21,22,32)(10,25,11,23)(13,27,26,31)(15,29,20,28)(34,41)(36,51)(37,38)(40,54)(42,58)(43,44)(45,56)(46,61)(47,48)(50,57)(55,59)(60,64);(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)(33,36,39,51)(34,42,44,50)(35,46,49,61)(37,52,38,53)(40,55,56,64)(41,57,43,58)(45,59,54,60)(47,62,48,63);(36,51)(42,50)(46,61)(52,53)(55,64)(57,58)(59,60)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 6258262);