Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367198 |
| 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 | |
1496 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)(3,13,14,24)(4,9,20,27)(5,11,6,10)(8,16,22,17)(15,29,30,25)(18,32,31,23)(19,28,21,26)(33,34,39,44)(35,40,48,55)(36,49,51,56)(37,42,38,43)(41,52,57,50)(45,59,62,63)(46,54,47,58)(53,61,64,60);(1,3,7,14)(2,8,12,22)(4,18,20,31)(5,17,6,16)(9,25,27,29)(10,13,11,24)(15,21,30,19)(23,28,32,26)(33,35,39,48)(34,40,44,55)(36,45,51,62)(37,46,38,47)(41,53,57,64)(42,54,43,58)(49,59,56,63)(50,60,52,61);(3,14)(8,22)(13,24)(15,30)(16,17)(18,31)(23,32)(25,29)(33,36)(34,41)(35,45)(37,50)(38,52)(39,51)(40,53)(42,56)(43,49)(44,57)(46,60)(47,61)(48,62)(54,63)(55,64)(58,59);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,23)(10,26)(11,28)(12,27)(13,29)(14,30)(16,18)(17,31)(22,32)(24,25)(33,37,39,38)(34,42,44,43)(35,46,48,47)(36,50,51,52)(40,54,55,58)(41,56,57,49)(45,60,62,61)(53,63,64,59);(1,5,7,6)(2,10,12,11)(3,16,14,17)(4,19,20,21)(8,24,22,13)(9,26,27,28)(15,18,30,31)(23,25,32,29)(33,37,39,38)(34,43,44,42)(35,47,48,46)(36,50,51,52)(40,54,55,58)(41,49,57,56)(45,61,62,60)(53,63,64,59)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274718);