Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357480 |
| 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 | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1759 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,30)(19,23)(21,27)(28,31)(29,32)(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);(1,3,6,14,7,15,5,19)(2,8,10,12,11,23,20,22)(4,13,18,27,17,28,16,29)(9,21,24,26,25,32,30,31)(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)(2,9)(3,13)(5,16)(6,18)(7,17)(8,21)(10,24)(11,25)(12,26)(14,27)(15,28)(19,29)(20,30)(22,31)(23,32)(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);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,16,17,18)(8,22,23,12)(9,24,25,30)(13,27,28,29)(21,31,32,26)(35,44)(39,52)(43,58)(45,57)(51,63)(53,60)(56,64)(59,62);(4,17)(9,25)(13,28)(16,18)(21,32)(24,30)(26,31)(27,29)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264698);