Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365522 |
| 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 | #3184 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,11,12,27)(4,8,17,25)(5,18,21,10)(7,19,22,14)(13,29,30,24)(15,32,31,23)(16,28,20,26)(33,34)(35,39)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57)(44,53)(45,61)(48,56)(50,55)(52,59)(54,60)(58,64)(62,63);(1,3,6,12)(2,7,9,22)(4,15,17,31)(5,19,21,14)(8,24,25,29)(10,27,18,11)(13,20,30,16)(23,28,32,26)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,53)(42,54)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(3,12)(7,22)(11,27)(13,30)(14,19)(15,31)(23,32)(24,29)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(42,56,49,46)(45,62,60,59)(54,64,61,57);(1,4,6,17)(2,8,9,25)(3,13,12,30)(5,20,21,16)(7,23,22,32)(10,28,18,26)(11,29,27,24)(14,31,19,15)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64);(2,9)(7,22)(8,25)(10,18)(11,27)(23,32)(24,29)(26,28)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272771);