Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368694 |
| 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 | |
1466 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)(3,13,17,30)(4,9,20,28)(5,11,6,10)(8,22,25,16)(14,24,18,32)(15,23,31,26)(19,29,21,27)(33,34)(35,41)(36,47)(37,42)(38,51)(39,45)(40,49)(43,50)(44,59)(46,54)(48,58)(52,56)(53,63)(55,61)(57,62)(60,64);(1,3)(2,8)(4,18)(5,16)(6,22)(7,17)(9,26)(10,13)(11,30)(12,25)(14,20)(15,19)(21,31)(23,28)(24,27)(29,32)(33,35,37,45)(34,39,42,41)(36,44,48,60)(38,46,50,62)(40,53,56,55)(43,54,51,57)(47,64,58,59)(49,61,52,63);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,23)(10,29)(11,27)(12,28)(13,24)(15,16)(17,18)(22,31)(25,26)(30,32)(33,36,37,48)(34,40,42,56)(35,44,45,60)(38,52,50,49)(39,53,41,55)(43,58,51,47)(46,63,62,61)(54,59,57,64);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,24)(10,27)(11,29)(12,28)(13,23)(14,16)(17,31)(18,22)(25,32)(26,30)(33,35)(34,41)(36,44)(37,45)(38,46)(39,42)(40,55)(43,57)(47,59)(48,60)(49,61)(50,62)(51,54)(52,63)(53,56)(58,64);(1,5,7,6)(2,10,12,11)(3,16,17,22)(4,19,20,21)(8,13,25,30)(9,27,28,29)(14,31,18,15)(23,32,26,24)(35,45)(39,41)(44,60)(46,62)(53,55)(54,57)(59,64)(61,63)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275955);