Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| 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 | #62190 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,15,22)(4,9,6,19)(5,10,18,27)(8,20,24,13)(14,29,31,26)(16,25,21,28)(17,32,30,23)(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)(2,8)(4,13)(5,17)(6,20)(7,15)(9,22)(10,26)(11,24)(12,19)(14,16)(18,30)(21,31)(23,28)(25,32)(27,29)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,18,21)(8,22,24,12)(10,25,27,28)(14,30,31,17)(23,26,32,29)(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);(1,5,7,18)(2,10,11,27)(3,14,15,31)(4,16,6,21)(8,23,24,32)(9,25,19,28)(12,29,22,26)(13,30,20,17)(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);(2,11)(3,15)(9,19)(10,27)(13,20)(14,31)(17,30)(25,28)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271619);