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:
(1,2,7,12)(3,13,18,24)(4,9,20,27)(5,11,6,10)(8,16,25,17)(14,29,32,23)(15,30,31,22)(19,28,21,26)(33,34,39,43)(35,40,46,55)(36,48,50,56)(37,42,38,51)(41,52,57,49)(44,58,60,63)(45,54,47,61)(53,62,64,59);(1,3,7,18)(2,8,12,25)(4,14,20,32)(5,17,6,16)(9,22,27,30)(10,13,11,24)(15,21,31,19)(23,28,29,26)(33,35,39,46)(34,40,43,55)(36,44,50,60)(37,45,38,47)(41,53,57,64)(42,54,51,61)(48,58,56,63)(49,59,52,62);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,28)(11,26)(12,27)(13,29)(15,16)(17,31)(18,32)(23,24)(25,30)(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);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,23)(10,26)(11,28)(12,27)(13,30)(14,16)(17,32)(18,31)(22,24)(25,29)(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);(1,5,7,6)(2,10,12,11)(3,16,18,17)(4,19,20,21)(8,24,25,13)(9,26,27,28)(14,31,32,15)(22,29,30,23)(34,43)(35,46)(41,57)(42,51)(44,60)(45,47)(48,56)(59,62)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,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, 6275613);