Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369070 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,18,30)(4,5,10,23)(6,12,7,11)(9,25,29,17)(15,16,19,32)(20,26,24,22)(21,31,27,28)(33,34)(35,40)(36,42)(37,44)(38,43)(39,45)(41,49)(46,54)(47,59)(48,56)(50,58)(51,55)(52,57)(53,61)(60,64)(62,63);(1,3,8,18)(2,9,13,29)(4,19,10,15)(5,21,23,27)(6,17,7,25)(11,14,12,30)(16,26,32,22)(20,31,24,28)(33,35,39,48)(34,40,45,56)(36,46,50,62)(37,51,38,47)(41,53,57,64)(42,54,58,63)(43,59,44,55)(49,61,52,60);(1,4,8,10)(2,5,13,23)(3,15,18,19)(6,24,7,20)(9,27,29,21)(11,26,12,22)(14,16,30,32)(17,31,25,28)(33,36)(34,41)(35,46)(37,49)(38,52)(39,50)(40,53)(42,43)(44,58)(45,57)(47,60)(48,62)(51,61)(54,59)(55,63)(56,64);(1,5)(2,10)(3,16)(4,13)(6,22)(7,26)(8,23)(9,28)(11,24)(12,20)(14,19)(15,30)(17,27)(18,32)(21,25)(29,31)(33,36)(34,42)(35,46)(37,52)(38,49)(39,50)(40,54)(41,43)(44,57)(45,58)(47,61)(48,62)(51,60)(53,59)(55,64)(56,63);(1,6,8,7)(2,11,13,12)(3,17,18,25)(4,20,10,24)(5,22,23,26)(9,14,29,30)(15,28,19,31)(16,27,32,21)(33,37,39,38)(34,43,45,44)(35,47,48,51)(36,49,50,52)(40,55,56,59)(41,42,57,58)(46,60,62,61)(53,63,64,54)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276341);