Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366870 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62145 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,22)(4,9,6,11)(5,10,20,28)(8,16,24,14)(15,29,31,26)(18,25,21,27)(19,32,30,23)(33,34,39,44)(35,40,48,55)(36,49,51,56)(37,42,38,43)(41,52,57,50)(45,59,62,63)(46,54,47,58)(53,61,64,60);(1,3,7,17)(2,8,12,24)(4,14,6,16)(5,19,20,30)(9,22,11,13)(10,26,28,29)(15,21,31,18)(23,25,32,27)(33,35,39,48)(34,40,44,55)(36,45,51,62)(37,46,38,47)(41,53,57,64)(42,54,43,58)(49,59,56,63)(50,60,52,61);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,20,21)(8,22,24,13)(10,25,28,27)(15,30,31,19)(23,26,32,29)(33,36)(34,41)(35,45)(37,50)(38,52)(39,51)(40,53)(42,56)(43,49)(44,57)(46,60)(47,61)(48,62)(54,63)(55,64)(58,59);(1,5,7,20)(2,10,12,28)(3,15,17,31)(4,18,6,21)(8,23,24,32)(9,25,11,27)(13,29,22,26)(14,30,16,19)(33,37,39,38)(34,42,44,43)(35,46,48,47)(36,50,51,52)(40,54,55,58)(41,56,57,49)(45,60,62,61)(53,63,64,59);(1,4,7,6)(2,11,12,9)(3,16,17,14)(5,18,20,21)(8,22,24,13)(10,27,28,25)(15,19,31,30)(23,26,32,29)(33,37,39,38)(34,43,44,42)(35,47,48,46)(36,50,51,52)(40,54,55,58)(41,49,57,56)(45,61,62,60)(53,63,64,59)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274383);