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 |
| prime radical | #62831 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,22)(4,9,20,26)(5,11,6,10)(8,16,23,15)(14,27,30,31)(18,25,19,24)(21,28,32,29)(33,34,39,42)(35,45,47,55)(36,50,51,58)(37,44,38,43)(40,49,56,48)(41,52,57,53)(46,60,61,54)(59,62,64,63);(1,3,7,17)(2,8,12,23)(4,14,20,30)(5,15,6,16)(9,21,26,32)(10,13,11,22)(18,28,19,29)(24,31,25,27)(33,35,39,47)(34,40,42,56)(36,46,51,61)(37,49,38,48)(41,54,57,60)(43,55,44,45)(50,64,58,59)(52,63,53,62);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,21)(10,25)(11,24)(12,26)(13,27)(15,29)(16,28)(17,30)(22,31)(23,32)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,18,20,19)(8,13,23,22)(9,24,26,25)(14,28,30,29)(21,31,32,27)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63);(1,5,7,6)(2,10,12,11)(3,16,17,15)(4,19,20,18)(8,22,23,13)(9,25,26,24)(14,28,30,29)(21,31,32,27)(34,42)(35,47)(41,57)(43,44)(46,61)(48,49)(50,58)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=hi,e^2=hi,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, 6256956);