Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| 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 | #3482 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1458 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)(3,12,13,28)(4,9,19,27)(5,10,6,21)(8,15,22,16)(14,29,17,32)(18,25,20,26)(23,31,24,30)(33,34,39,44)(35,45,49,56)(36,50,53,59)(37,42,38,43)(40,47,57,48)(41,54,60,52)(46,58,51,55)(61,62,64,63);(1,3,6,16,7,13,5,15)(2,8,10,12,11,22,21,28)(4,17,18,31,19,14,20,30)(9,24,26,32,27,23,25,29)(33,35,38,47,39,49,37,48)(34,40,42,56,44,57,43,45)(36,51,54,63,53,46,52,62)(41,58,59,61,60,55,50,64);(3,13)(8,22)(12,28)(14,17)(15,16)(23,24)(29,32)(30,31)(33,36)(34,41)(35,46)(37,52)(38,54)(39,53)(40,55)(42,59)(43,50)(44,60)(45,61)(47,62)(48,63)(49,51)(56,64)(57,58);(1,4)(2,9)(3,14)(5,20)(6,18)(7,19)(8,23)(10,26)(11,27)(12,29)(13,17)(15,31)(16,30)(21,25)(22,24)(28,32)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,52,53,54)(40,45,57,56)(41,59,60,50)(46,62,51,63)(55,61,58,64);(1,5,7,6)(2,10,11,21)(3,15,13,16)(4,18,19,20)(8,12,22,28)(9,25,27,26)(14,30,17,31)(23,32,24,29)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264213);