Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1498 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)(3,8,13,25)(4,9,17,27)(5,18,21,11)(7,19,23,14)(12,24,30,26)(15,32,31,22)(16,29,20,28)(33,34,39,43)(35,40,46,55)(36,48,50,56)(37,42,38,51)(41,52,57,49)(44,58,60,63)(45,54,47,61)(53,62,64,59);(1,3)(2,7)(4,15)(5,19)(6,13)(8,18)(9,26)(10,23)(11,25)(12,16)(14,21)(17,31)(20,30)(22,28)(24,27)(29,32)(33,35,39,46)(34,40,43,55)(36,44,50,60)(37,45,38,47)(41,53,57,64)(42,54,51,61)(48,58,56,63)(49,59,52,62);(1,3)(2,8)(4,12)(5,14)(6,13)(7,18)(9,24)(10,25)(11,23)(15,16)(17,30)(19,21)(20,31)(22,28)(26,27)(29,32)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,59)(46,60)(47,62)(48,51)(54,63)(55,64)(58,61);(1,4,6,17)(2,9,10,27)(3,12,13,30)(5,20,21,16)(7,22,23,32)(8,24,25,26)(11,29,18,28)(14,31,19,15)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(2,10)(3,13)(9,27)(11,18)(12,30)(14,19)(15,31)(28,29)(34,43)(35,46)(41,57)(42,51)(44,60)(45,47)(48,56)(59,62)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^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,e^c=eh,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, 6275603);