Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235492 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #37543 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14)(3,10,17,41)(4,8,21,11)(5,9,24,34)(7,15,18,13)(12,26,35,25)(16,42,43,40)(19,28,46,38)(20,33,47,37)(22,31,48,30)(23,54,52,32)(27,55,29,53)(36,49,58,44)(39,45,60,51)(50,61,63,59)(56,64,57,62);(2,8)(3,16)(4,21)(5,24)(9,32)(10,36)(11,14)(12,27)(13,30)(15,31)(17,43)(19,49)(20,51)(22,48)(25,26)(28,42)(29,35)(33,57)(34,54)(37,56)(38,40)(39,61)(41,58)(44,46)(45,47)(50,62)(59,60)(63,64);(2,9)(3,17)(4,22)(5,25)(8,27)(10,37)(11,29)(12,13)(14,34)(15,35)(16,43)(19,49)(20,45)(21,48)(23,52)(24,26)(28,57)(30,32)(31,54)(33,41)(36,59)(38,56)(39,42)(40,60)(44,46)(47,51)(53,55)(58,61);(1,3,18,43)(2,10,13,40)(4,19,48,49)(5,20,25,51)(6,17,7,16)(8,28,30,58)(9,33,12,39)(11,38,31,36)(14,41,15,42)(21,46,22,44)(23,50,55,64)(24,47,26,45)(27,56,32,59)(29,57,54,61)(34,37,35,60)(52,63,53,62)Polycyclic presentation:
2^9|a^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142699);