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 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,4)(3,11,15,27)(5,9,20,16)(6,17,23,10)(8,21,12,14)(13,29,31,25)(18,30,32,24)(19,28,22,26)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,58)(45,54)(46,55)(47,61)(50,56)(52,57)(53,59)(60,63)(62,64);(1,3,7,15)(2,8,4,12)(5,18,20,32)(6,21,23,14)(9,25,16,29)(10,27,17,11)(13,22,31,19)(24,28,30,26)(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);(1,4,7,2)(3,12,15,8)(5,16,20,9)(6,17,23,10)(11,14,27,21)(13,30,31,24)(18,29,32,25)(19,28,22,26)(33,36,39,50)(34,41,43,57)(35,44,46,60)(37,49,38,52)(40,53,55,64)(42,56,51,48)(45,59,47,62)(54,63,61,58);(1,5,7,20)(2,9,4,16)(3,13,15,31)(6,22,23,19)(8,24,12,30)(10,28,17,26)(11,29,27,25)(14,32,21,18)(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);(2,4)(8,12)(9,16)(10,17)(11,27)(24,30)(25,29)(26,28)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,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=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, 6272065);