Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187410 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1980 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)(3,13,17,8)(4,18,21,25)(5,10)(6,12)(7,11)(9,20,26,19)(14,28)(15,24,16,23)(22,30)(27,29)(31,32)(33,34,38,43,39,44,37,42)(35,45,49,40,48,55,47,54)(36,41,50,56,51,57,52,58)(46,60,61,64,62,59,63,53);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,32)(19,29)(20,30)(21,31)(25,28)(26,27)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,55)(43,45)(44,54)(50,63)(51,62)(52,61)(56,59)(57,64)(58,60);(1,4,7,21)(2,9,11,26)(3,14,17,31)(5,20,6,19)(8,22,13,27)(10,18,12,25)(15,30,16,29)(23,32,24,28)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,58)(43,56)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,21,20)(8,23,13,24)(9,25,26,18)(14,29,31,30)(22,28,27,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94721);