Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369150 |
| 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 | #62378 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(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)(33,34,37,42)(35,41,45,39)(36,47,49,59)(38,52,51,43)(40,53,57,50)(44,60,48,64)(46,55,62,58)(54,61,56,63);(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)(33,35,37,45)(34,39,42,41)(36,48,49,44)(38,46,51,62)(40,56,57,54)(43,55,52,58)(47,60,59,64)(50,63,53,61);(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)(33,36,37,49)(34,40,42,57)(35,44,45,48)(38,53,51,50)(39,54,41,56)(43,59,52,47)(46,63,62,61)(55,60,58,64);(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)(33,35)(34,41)(36,44)(37,45)(38,46)(39,42)(40,56)(43,58)(47,60)(48,49)(50,61)(51,62)(52,55)(53,63)(54,57)(59,64);(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)(34,42)(35,45)(40,57)(43,52)(44,48)(46,62)(47,59)(61,63)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276392);