Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368046 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1479 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)(3,14,19,27)(4,5,10,23)(6,12,7,11)(9,17,28,18)(15,30,29,16)(20,25,24,22)(21,31,32,26)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,58)(45,54)(46,55)(47,61)(50,56)(52,57)(53,59)(60,63)(62,64);(1,3,8,19)(2,9,13,28)(4,15,10,29)(5,21,23,32)(6,18,7,17)(11,14,12,27)(16,25,30,22)(20,31,24,26)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(1,4,8,10)(2,5,13,23)(3,15,19,29)(6,24,7,20)(9,21,28,32)(11,25,12,22)(14,30,27,16)(17,31,18,26)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,59)(46,60)(47,62)(48,51)(54,63)(55,64)(58,61);(1,5)(2,10)(3,16)(4,13)(6,22)(7,25)(8,23)(9,26)(11,24)(12,20)(14,15)(17,21)(18,32)(19,30)(27,29)(28,31)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(1,6,8,7)(2,11,13,12)(3,17,19,18)(4,20,10,24)(5,22,23,25)(9,27,28,14)(15,26,29,31)(16,21,30,32)(34,43)(35,46)(41,57)(42,51)(44,60)(45,47)(48,56)(59,62)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275521);