Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349684 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,11)(4,8)(5,17)(6,9)(7,14)(10,19)(12,25)(13,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31)(33,34)(35,40)(36,48)(37,43)(38,42)(39,44)(41,49)(45,59)(46,55)(47,56)(50,57)(51,54)(52,58)(53,62)(60,64)(61,63);(1,3)(2,7)(4,12)(5,18)(6,13)(8,20)(9,21)(10,24)(11,17)(14,19)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28)(33,35)(34,40)(36,45)(37,46)(38,51)(39,47)(41,53)(42,54)(43,55)(44,56)(48,63)(49,60)(50,61)(52,62)(57,59)(58,64);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,22)(10,23)(11,25)(13,26)(14,27)(17,29)(18,30)(19,28)(21,31)(24,32)(33,36)(34,41)(35,45)(37,49)(38,52)(39,50)(40,53)(42,48)(43,57)(44,58)(46,60)(47,61)(51,62)(54,63)(55,59)(56,64);(2,9)(4,15)(7,21)(10,17)(11,24)(12,26)(16,28)(27,30)(33,37,39,38)(34,42,44,43)(35,46,47,51)(36,49,50,52)(40,54,56,55)(41,48,58,57)(45,60,61,62)(53,63,64,59);(2,9)(3,13)(8,22)(10,17)(12,26)(14,18)(23,29)(27,30)(33,37,39,38)(34,43,44,42)(35,46,47,51)(36,49,50,52)(40,55,56,54)(41,57,58,48)(45,60,61,62)(53,59,64,63)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256944);