Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6408568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #34364 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60651 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,14,18,8)(4,19)(5,24,26,11)(6,10)(7,12)(9,23)(13,28)(15,41,42,33)(16,31)(17,32,27,30)(20,39,47,29)(21,36)(22,51,49,46)(25,38,55,37)(34,52,53,64)(35,50)(40,45)(43,59)(44,57)(48,56)(54,58)(60,63)(61,62);(1,3)(2,8)(4,20)(5,16)(6,27)(7,18)(9,33)(10,32)(11,31)(12,14)(13,30)(15,23)(17,28)(19,29)(21,47)(22,48)(24,40)(25,54)(26,45)(34,59)(35,41)(36,39)(37,58)(38,57)(42,50)(43,52)(44,55)(46,56)(49,63)(51,60)(53,61)(62,64);(1,4,6,23,7,21,28,50)(2,9,13,19,12,35,10,36)(3,15,17,20,18,42,27,47)(5,22,25,52,26,49,55,64)(8,29,32,33,14,39,30,41)(11,34,38,46,24,53,37,51)(16,43,44,48,45,62,54,63)(31,56,58,59,40,60,57,61);(2,10,12,13)(4,21)(8,30,14,32)(9,19,35,36)(11,37,24,38)(15,42)(20,47)(22,49)(23,50)(29,33,39,41)(31,57,40,58)(34,46,53,51)(43,62)(48,63)(52,64)(56,59,60,61);(1,5,7,26)(2,11,12,24)(3,16,18,45)(4,22,21,49)(6,25,28,55)(8,31,14,40)(9,34,35,53)(10,37,13,38)(15,43,42,62)(17,44,27,54)(19,46,36,51)(20,48,47,63)(23,52,50,64)(29,56,39,60)(30,57,32,58)(33,59,41,61)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,c^b=cg,d^b=df,g^b=gi,h^b=hi,d^c=di,h^e=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315765);