Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,19,28)(4,10,22,5)(6,12,7,11)(9,17,29,18)(15,31,20,16)(21,24,25,26)(23,32,30,27)(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);(1,3,8,19)(2,9,13,29)(4,20,22,15)(5,23,10,30)(6,18,7,17)(11,14,12,28)(16,26,31,24)(21,27,25,32)(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);(1,4)(2,10)(3,15)(5,13)(6,25)(7,21)(8,22)(9,23)(11,24)(12,26)(14,31)(16,28)(17,32)(18,27)(19,20)(29,30)(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);(1,5,8,10)(2,4,13,22)(3,16,19,31)(6,24,7,26)(9,27,29,32)(11,21,12,25)(14,15,28,20)(17,23,18,30);(1,6,8,7)(2,11,13,12)(3,17,19,18)(4,21,22,25)(5,24,10,26)(9,28,29,14)(15,27,20,32)(16,23,31,30)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272575);