Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214040 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
724 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)(3,12,15,7)(4,16,18,9)(5,11)(6,10)(8,17,26,20)(13,23)(14,28,19,24)(21,27)(22,30)(25,32)(29,31)(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);(1,3,6,15)(2,7,10,12)(4,13,18,31)(5,19,21,14)(8,25,26,22)(9,23,16,29)(11,28,27,24)(17,32,20,30)(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)(2,8)(3,13)(5,20)(6,18)(7,22)(9,11)(10,26)(12,25)(14,32)(15,31)(16,27)(17,21)(19,30)(23,24)(28,29)(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);(1,4)(2,9)(3,13)(5,17)(6,18)(7,23)(8,11)(10,16)(12,29)(14,30)(15,31)(19,32)(20,21)(22,24)(25,28)(26,27)(35,44)(39,52)(43,58)(45,60)(51,57)(53,63)(59,61)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,f^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,f^c=fh,g^c=ghi,g^d=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121243);