Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
774 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)(3,11,14,7)(4,8)(5,17,18,9)(6,10)(12,19,26,21)(13,20)(15,28,29,22)(16,23)(24,27)(25,31)(30,32)(33,34,37,42,38,41,50,55)(35,39,45,53,44,52,60,63)(36,46,47,56,48,40,49,54)(43,57,59,64,58,51,61,62);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,20)(10,11)(15,25)(16,26)(17,24)(18,27)(19,23)(22,31)(28,30)(29,32)(33,35)(34,39)(36,43)(37,45)(38,44)(40,51)(41,52)(42,53)(46,57)(47,59)(48,58)(49,61)(50,60)(54,62)(55,63)(56,64);(1,4,6,16)(2,8,10,23)(3,12,14,26)(5,15,18,29)(7,19,11,21)(9,22,17,28)(13,25,27,32)(20,30,24,31)(33,36,38,48)(34,40,41,46)(35,43,44,58)(37,49,50,47)(39,51,52,57)(42,56,55,54)(45,61,60,59)(53,64,63,62);(1,5,6,18)(2,9,10,17)(3,13,14,27)(4,15,16,29)(7,20,11,24)(8,22,23,28)(12,25,26,32)(19,30,21,31)(35,44)(39,52)(43,58)(45,60)(51,57)(53,63)(59,61)(62,64)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121187);