Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350061 |
| 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 | #63036 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2606 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)(6,19)(7,11)(8,13)(14,27)(15,21)(16,24)(17,25)(18,26)(20,23)(22,28)(29,32)(30,31)(33,34,39,42)(35,40,46,54)(36,47,49,58)(37,43,38,50)(41,52,57,48)(44,59,61,63)(45,55,51,56)(53,60,64,62);(1,3,7,15)(2,8,11,23)(4,13,6,20)(5,14,18,30)(9,21,19,12)(10,22,26,32)(16,28,17,29)(24,31,25,27)(33,35,39,46)(34,40,42,54)(36,44,49,61)(37,45,38,51)(41,53,57,64)(43,55,50,56)(47,63,58,59)(48,60,52,62);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,16,18,17)(8,21,23,12)(10,24,26,25)(14,28,30,29)(22,31,32,27)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,61)(47,50)(51,62)(54,64)(55,59)(56,63);(1,5,7,18)(2,10,11,26)(3,14,15,30)(4,16,6,17)(8,22,23,32)(9,24,19,25)(12,27,21,31)(13,28,20,29)(34,42)(40,54)(41,57)(43,50)(47,58)(53,64)(55,56)(59,63);(1,4,7,6)(2,9,11,19)(3,13,15,20)(5,17,18,16)(8,21,23,12)(10,25,26,24)(14,29,30,28)(22,27,32,31)(33,37,39,38)(34,43,42,50)(35,45,46,51)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,61,62)(53,59,64,63)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257559);