Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,5,10)(3,13,17,24,16,23,15,8)(4,18,20,25,21,9,19,26)(14,27,30,22,29,32,28,31)(33,34)(35,43,46,57)(36,47,49,40)(37,42)(38,41)(39,45,54,50)(44,59)(48,58,51,55)(52,56)(53,60)(61,64)(62,63);(1,3,7,16)(2,8)(4,14,21,29)(5,15,6,17)(9,22)(10,23)(11,13)(12,24)(18,31)(19,28,20,30)(25,27)(26,32)(33,35,37,50,38,46,52,45)(34,39,42,57,41,54,56,43)(36,44,51,60,49,61,48,62)(40,53,58,64,47,63,55,59);(1,4,7,21)(2,9,12,18)(3,14,16,29)(5,20,6,19)(8,22,24,31)(10,26,11,25)(13,27,23,32)(15,30,17,28)(33,36)(34,40)(35,44)(37,51)(38,49)(39,53)(41,47)(42,58)(43,59)(45,62)(46,61)(48,52)(50,60)(54,63)(55,56)(57,64);(1,5,7,6)(2,10,12,11)(3,15,16,17)(4,19,21,20)(8,23,24,13)(9,25,18,26)(14,28,29,30)(22,27,31,32)(34,41)(39,54)(40,47)(42,56)(43,57)(53,63)(55,58)(59,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108375);