Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356733 |
| 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 | |
1470 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)(3,12,16,29)(4,9,19,27)(5,10,6,21)(8,14,23,15)(13,28,17,32)(18,25,20,26)(22,31,24,30)(33,34,39,44)(35,45,49,55)(36,50,52,57)(37,42,38,43)(40,47,56,48)(41,53,58,51)(46,60,63,54)(59,61,64,62);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,21,29)(4,17,18,31,19,13,20,30)(9,24,26,32,27,22,25,28)(33,35,38,47,39,49,37,48)(34,40,42,55,44,56,43,45)(36,46,53,61,52,63,51,62)(41,54,57,64,58,60,50,59);(33,36)(34,41)(35,46)(37,51)(38,53)(39,52)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,62)(49,63)(55,64)(56,60);(1,4)(2,9)(3,13)(5,20)(6,18)(7,19)(8,22)(10,26)(11,27)(12,28)(14,31)(15,30)(16,17)(21,25)(23,24)(29,32)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,45,56,55)(41,57,58,50)(46,61,63,62)(54,59,60,64);(1,5,7,6)(2,10,11,21)(3,14,16,15)(4,18,19,20)(8,12,23,29)(9,25,27,26)(13,30,17,31)(22,32,24,28)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,d^b=dh,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, 6264045);