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 | #62967 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(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)(33,34,39,45)(35,42,49,56)(36,50,54,59)(37,44,38,43)(40,47,57,48)(41,53,60,52)(46,62,51,55)(58,64,61,63);(1,3,7,16)(2,8,12,22)(4,14,17,28)(5,15,6,20)(9,21,23,31)(10,13,11,26)(18,29,19,30)(24,27,25,32)(33,35)(34,40)(36,46)(37,47)(38,48)(39,49)(41,55)(42,44)(43,56)(45,57)(50,58)(51,54)(52,63)(53,64)(59,61)(60,62);(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)(34,41)(35,46)(37,52)(38,53)(39,54)(40,55)(42,61)(43,50)(44,59)(45,60)(47,63)(48,64)(49,51)(56,58)(57,62);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(33,35)(34,42)(36,51)(37,48)(38,47)(39,49)(40,44)(41,58)(43,57)(45,56)(46,54)(50,55)(52,63)(53,64)(59,62)(60,61);(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)(33,37,39,38)(34,43,45,44)(35,47,49,48)(36,52,54,53)(40,56,57,42)(41,50,60,59)(46,63,51,64)(55,58,62,61)Polycyclic presentation:
2^9|a^2=h,b^2=hi,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, 6258380);