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 |
| prime radical | #3554 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1440 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)(3,11)(4,7)(5,9)(6,17)(8,13)(10,23)(12,29)(14,27)(15,21)(16,20)(18,32)(19,28)(22,26)(24,31)(25,30)(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);(1,3,7,14)(2,8,4,15)(5,18,20,31)(6,21,23,13)(9,25,16,29)(10,27,17,11)(12,22,30,19)(24,28,32,26)(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);(1,4,7,2)(3,8,14,15)(5,16,20,9)(6,17,23,10)(11,21,27,13)(12,24,30,32)(18,25,31,29)(19,28,22,26)(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);(1,5,7,20)(2,9,4,16)(3,12,14,30)(6,22,23,19)(8,24,15,32)(10,28,17,26)(11,29,27,25)(13,31,21,18)(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);(2,4)(8,15)(9,16)(10,17)(11,27)(24,32)(25,29)(26,28)(35,45)(39,41)(44,60)(46,62)(53,55)(54,57)(59,64)(61,63)Polycyclic presentation:
2^9|b^2=hi,c^2=hi,d^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, 6275953);