Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347034 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,10)(3,16)(4,21)(5,23)(6,22)(11,38)(12,31)(13,46)(14,26)(15,37)(17,27)(18,52)(19,32)(20,50)(24,25)(28,33)(29,30)(34,48)(35,59)(36,41)(39,43)(40,47)(42,62)(44,55)(45,58)(49,51)(53,56)(54,61)(57,63)(60,64);(1,2,8,14)(3,11,20,42)(4,12,24,45)(5,26,23,10)(6,31,28,58)(7,34,9,15)(13,33,44,22)(16,47,50,61)(17,39,53,64)(18,54,51,40)(19,43,35,60)(21,55,25,46)(27,57,56,36)(29,37,30,48)(32,41,59,63)(38,49,62,52);(1,3)(2,11)(4,17)(5,27)(6,32)(7,35)(8,20)(9,19)(10,36)(12,39)(13,47)(14,42)(15,43)(16,22)(18,30)(21,49)(23,56)(24,53)(25,52)(26,57)(28,59)(29,51)(31,41)(33,50)(34,60)(37,40)(38,46)(44,61)(45,64)(48,54)(55,62)(58,63);(1,4,8,24)(2,12,14,45)(3,17,20,53)(5,28,23,6)(7,21,9,25)(10,31,26,58)(11,39,42,64)(13,48,44,37)(15,46,34,55)(16,18,50,51)(19,52,35,49)(22,30,33,29)(27,59,56,32)(36,41,57,63)(38,60,62,43)(40,47,54,61)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gh,e^d=ehi,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254235);