Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364762 |
| 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 | |
1487 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)(3,13,17,28)(4,9,19,26)(5,11,6,10)(8,21,24,16)(14,29,32,23)(15,30,31,22)(18,27,20,25)(33,34)(35,40)(36,47)(37,42)(38,51)(39,43)(41,48)(44,59)(45,54)(46,56)(49,57)(50,55)(52,58)(53,61)(60,63)(62,64);(1,3,7,17)(2,8,12,24)(4,14,19,32)(5,16,6,21)(9,22,26,30)(10,13,11,28)(15,20,31,18)(23,27,29,25)(33,35,39,46)(34,40,43,56)(36,44,49,62)(37,50,38,45)(41,53,58,63)(42,55,51,54)(47,59,57,64)(48,61,52,60);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,22)(10,27)(11,25)(12,26)(13,29)(15,21)(16,31)(17,32)(23,28)(24,30)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,62)(47,51)(50,61)(54,59)(55,64)(56,63);(1,4)(2,9)(3,15)(5,18)(6,20)(7,19)(8,23)(10,25)(11,27)(12,26)(13,30)(14,21)(16,32)(17,31)(22,28)(24,29);(1,5,7,6)(2,10,12,11)(3,16,17,21)(4,18,19,20)(8,13,24,28)(9,25,26,27)(14,15,32,31)(22,23,30,29)(33,37,39,38)(34,42,43,51)(35,45,46,50)(36,48,49,52)(40,54,56,55)(41,57,58,47)(44,60,62,61)(53,59,63,64)Polycyclic presentation:
2^9|a^2=h,b^2=hi,e^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=ei,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, 6272136);