Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355925 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3480 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,13,26)(4,9,18,25)(5,10,6,20)(8,15,21,16)(14,27,28,32)(17,23,19,24)(22,30,31,29)(33,34,39,44)(35,40,46,54)(36,49,51,58)(37,43,38,42)(41,52,57,50)(45,59,60,64)(47,56,48,55)(53,62,63,61);(1,3,5,15,7,13,6,16)(2,8,20,26,11,21,10,12)(4,14,19,30,18,28,17,29)(9,22,23,32,25,31,24,27)(33,35)(34,40)(36,45)(37,47)(38,48)(39,46)(41,53)(42,55)(43,56)(44,54)(49,59)(50,61)(51,60)(52,62)(57,63)(58,64);(3,13)(8,21)(12,26)(14,28)(15,16)(22,31)(27,32)(29,30)(33,36,39,51)(34,41,44,57)(35,45,46,60)(37,50,38,52)(40,53,54,63)(42,49,43,58)(47,61,48,62)(55,59,56,64);(1,4)(2,9)(3,14)(5,19)(6,17)(7,18)(8,22)(10,24)(11,25)(12,27)(13,28)(15,30)(16,29)(20,23)(21,31)(26,32)(35,46)(40,54)(45,60)(47,48)(53,63)(55,56)(59,64)(61,62);(1,5,7,6)(2,10,11,20)(3,15,13,16)(4,17,18,19)(8,12,21,26)(9,23,25,24)(14,29,28,30)(22,32,31,27)(33,37,39,38)(34,42,44,43)(35,47,46,48)(36,50,51,52)(40,55,54,56)(41,49,57,58)(45,61,60,62)(53,59,63,64)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263192);