Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62953 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2542 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)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(33,34,40,46)(35,42,49,56)(36,41,48,57)(37,43,54,62)(38,45,39,44)(47,58,63,59)(50,55,51,64)(52,61,53,60);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,27)(25,32)(33,35,40,49)(34,41,46,57)(36,38,48,39)(37,47,54,63)(42,44,56,45)(43,55,62,64)(50,53,51,52)(58,61,59,60);(1,4)(2,9)(3,14)(5,18)(6,19)(7,17)(8,21)(10,24)(11,25)(12,23)(13,27)(15,29)(16,28)(20,30)(22,31)(26,32)(33,36)(34,42)(35,38)(37,51)(39,49)(40,48)(41,44)(43,59)(45,57)(46,56)(47,52)(50,54)(53,63)(55,60)(58,62)(61,64);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(33,37)(34,43)(35,47)(36,50)(38,53)(39,52)(40,54)(41,55)(42,58)(44,61)(45,60)(46,62)(48,51)(49,63)(56,59)(57,64);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,18,17,19)(8,13,22,26)(9,24,23,25)(14,29,28,30)(21,27,31,32)(33,38,40,39)(34,44,46,45)(35,48,49,36)(37,52,54,53)(41,56,57,42)(43,60,62,61)(47,50,63,51)(55,58,64,59)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258378);