Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368558 |
| 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 |
| prime radical | #62553 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,18,29)(4,5,10,22)(6,12,7,11)(9,24,27,17)(15,30,28,16)(19,25,23,21)(20,31,32,26)(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)(2,9)(4,15)(5,20)(6,17)(7,24)(8,18)(10,28)(11,14)(12,29)(13,27)(16,21)(19,31)(22,32)(23,26)(25,30)(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);(1,4,8,10)(2,5,13,22)(3,15,18,28)(6,23,7,19)(9,20,27,32)(11,25,12,21)(14,30,29,16)(17,26,24,31)(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);(1,5)(2,10)(3,16)(4,13)(6,21)(7,25)(8,22)(9,26)(11,23)(12,19)(14,15)(17,32)(18,30)(20,24)(27,31)(28,29)(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);(1,6,8,7)(2,11,13,12)(3,17,18,24)(4,19,10,23)(5,21,22,25)(9,14,27,29)(15,31,28,26)(16,32,30,20)(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=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275823);