Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355107 |
| 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 | C2×#41 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1456 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)(3,12,16,27)(4,9,18,25)(5,10,6,20)(8,14,22,15)(13,26,29,31)(17,23,19,24)(21,30,32,28)(33,34,38,42)(35,39,44,53)(36,46,48,56)(37,49,51,41)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(1,3,5,14,7,16,6,15)(2,8,20,27,11,22,10,12)(4,13,19,30,18,29,17,28)(9,21,23,31,25,32,24,26)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(33,36,38,48)(34,40,42,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(41,56,49,46)(45,62,60,59)(54,64,61,57);(1,4)(2,9)(3,13)(5,19)(6,17)(7,18)(8,21)(10,24)(11,25)(12,26)(14,30)(15,28)(16,29)(20,23)(22,32)(27,31)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,18,19)(8,12,22,27)(9,23,25,24)(13,28,29,30)(21,31,32,26)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,g^c=gi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262334);