Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356633 |
| 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 | #62129 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,13,24)(4,8,16,22)(5,18,12,9)(7,19,20,15)(14,25,27,32)(17,29,26,23)(21,30,31,28)(33,34,39,44)(35,45,49,56)(36,50,53,59)(37,42,38,43)(40,48,57,47)(41,54,60,52)(46,58,51,55)(61,63,64,62);(1,3,12,19,6,13,5,15)(2,7,18,24,10,20,9,11)(4,14,26,30,16,27,17,28)(8,21,29,32,22,31,23,25)(33,35,37,47,39,49,38,48)(34,40,43,45,44,57,42,56)(36,51,52,63,53,46,54,62)(41,58,50,64,60,55,59,61);(3,13)(7,20)(11,24)(14,27)(15,19)(21,31)(25,32)(28,30)(33,36,39,53)(34,41,44,60)(35,46,49,51)(37,52,38,54)(40,55,57,58)(42,59,43,50)(45,61,56,64)(47,62,48,63);(1,4,6,16)(2,8,10,22)(3,14,13,27)(5,17,12,26)(7,21,20,31)(9,23,18,29)(11,25,24,32)(15,28,19,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,52,53,54)(40,56,57,45)(41,59,60,50)(46,62,51,63)(55,64,58,61);(4,16)(8,22)(14,27)(17,26)(21,31)(23,29)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,g^c=gi,e^d=ehi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263925);