Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369070 |
| 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 | #466 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1436 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)(3,14,18,30)(4,5,10,23)(6,12,7,11)(9,25,29,17)(15,16,19,32)(20,26,24,22)(21,31,27,28)(33,34,37,42)(35,41,45,39)(36,47,48,58)(38,51,50,43)(40,52,56,49)(44,59,60,64)(46,54,62,57)(53,61,55,63);(1,3,8,18)(2,9,13,29)(4,19,10,15)(5,21,23,27)(6,17,7,25)(11,14,12,30)(16,26,32,22)(20,31,24,28)(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)(2,5,13,23)(3,15,18,19)(6,24,7,20)(9,27,29,21)(11,26,12,22)(14,16,30,32)(17,31,25,28)(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)(6,22)(7,26)(8,23)(9,28)(11,24)(12,20)(14,19)(15,30)(17,27)(18,32)(21,25)(29,31)(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)(3,17,18,25)(4,20,10,24)(5,22,23,26)(9,14,29,30)(15,28,19,31)(16,27,32,21)(34,42)(35,45)(40,56)(43,51)(44,60)(46,62)(47,58)(61,63)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276328);