Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×#121 |
| prime radical | #60953 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,11)(3,12,16,26)(4,9,17,23)(5,10,6,20)(8,14,22,15)(13,25,27,32)(18,24,19,30)(21,28,31,29)(33,34,39,44)(35,45,47,56)(36,50,51,58)(37,43,38,42)(40,49,55,48)(41,53,57,52)(46,60,62,54)(59,63,64,61);(1,3,6,15,7,16,5,14)(2,8,10,12,11,22,20,26)(4,13,19,29,17,27,18,28)(9,21,24,25,23,31,30,32)(33,35,37,48,39,47,38,49)(34,40,42,45,44,55,43,56)(36,46,52,61,51,62,53,63)(41,54,50,59,57,60,58,64);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,28)(15,29)(16,27)(20,30)(22,31)(26,32)(33,36)(34,41)(35,46)(37,52)(38,53)(39,51)(40,54)(42,50)(43,58)(44,57)(45,59)(47,62)(48,61)(49,63)(55,60)(56,64);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(25,32)(28,29)(35,47)(36,51)(40,55)(41,57)(45,56)(48,49)(50,58)(52,53);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,17,19)(8,12,22,26)(9,24,23,30)(13,28,27,29)(21,25,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261843);