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 | #121 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1443 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,13)(14,29)(15,22)(16,25)(17,32)(18,27)(20,24)(21,28)(23,30)(26,31)(33,34,38,42)(35,39,44,53)(36,46,48,56)(37,49,51,41)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(1,3,7,15)(2,8,11,24)(4,13,6,20)(5,17,18,30)(9,22,19,12)(10,26,27,29)(14,21,31,16)(23,25,32,28)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(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)(33,36,38,48)(34,40,42,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(41,56,49,46)(45,62,60,59)(54,64,61,57);(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);(2,11)(3,15)(9,19)(10,27)(13,20)(14,31)(17,30)(25,28)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=h,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, 6271556);