Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #250455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #61872 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,11)(4,15)(5,18)(6,10)(7,13)(8,16)(9,21)(12,23)(14,25)(17,26)(19,22)(20,24)(27,28)(29,31)(30,32)(33,34,39,43)(35,44,47,54)(36,49,51,58)(37,52,38,42)(40,48,56,45)(41,53,57,50)(46,55,62,64)(59,60,63,61);(1,3)(2,7)(4,12)(5,19)(6,14)(8,23)(9,25)(10,22)(11,18)(13,21)(15,27)(16,30)(17,29)(20,28)(24,31)(26,32)(33,35)(34,40)(36,46)(37,48)(38,45)(39,47)(41,55)(42,54)(43,56)(44,52)(49,59)(50,60)(51,62)(53,61)(57,64)(58,63);(7,22)(11,25)(23,31)(27,32)(35,45)(37,38)(40,54)(42,52)(44,56)(46,60)(47,48)(49,58)(50,53)(55,63)(59,64)(61,62);(1,4,6,17)(2,8,10,24)(3,12,14,29)(5,20,21,16)(7,23,22,31)(9,26,18,15)(11,27,25,32)(13,30,19,28)(33,36)(34,41)(35,46)(37,53)(38,50)(39,51)(40,55)(42,58)(43,57)(44,59)(45,60)(47,62)(48,61)(49,52)(54,63)(56,64)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,g^c=gi,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 157686);