Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357716 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3511 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,15,28)(4,8,16,24)(5,18,19,9)(7,20,23,14)(12,26,13,27)(17,32,29,25)(21,31,22,30)(33,34,38,42)(35,39,44,52)(36,46,49,61)(37,50,47,41)(40,54,55,48)(43,56,58,64)(45,60,57,53)(51,62,63,59);(1,3,5,14,6,15,19,20)(2,7,9,11,10,23,18,28)(4,13,17,31,16,12,29,30)(8,22,25,27,24,21,32,26)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,53)(42,52)(46,56)(47,57)(48,59)(49,58)(50,60)(54,62)(55,63)(61,64);(1,4)(2,8)(3,12)(5,17)(6,16)(7,21)(9,25)(10,24)(11,26)(13,15)(14,30)(18,32)(19,29)(20,31)(22,23)(27,28)(33,36,47,54,38,49,37,48)(34,40,50,61,42,55,41,46)(35,43,57,62,44,58,45,59)(39,51,60,64,52,63,53,56);(1,4)(2,8)(3,13)(5,17)(6,16)(7,22)(9,25)(10,24)(11,27)(12,15)(14,31)(18,32)(19,29)(20,30)(21,23)(26,28)(35,44)(39,52)(43,58)(45,57)(51,63)(53,60)(56,64)(59,62);(4,16)(8,24)(12,13)(17,29)(21,22)(25,32)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264937);