Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508561 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29942 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,16,9,14,6,13)(3,17,20,37,21,47,34,10)(4,22,27,41,28,11,25,43)(5,12,30,15,31,44,7,32)(18,50,52,63,53,48,56,35)(19,49,33,38,55,45,62,36)(23,59,60,39,51,64,54,57)(24,46,26,42,61,40,29,58);(1,3)(2,10,14,37)(4,23)(5,19)(6,20)(7,33)(8,34)(9,21)(11,39,22,57)(12,36,44,38)(13,17,16,47)(15,45,32,49)(18,26)(24,52)(25,60)(27,54)(28,51)(29,53)(30,62)(31,55)(35,58,63,42)(40,48,46,50)(41,64,43,59)(56,61);(1,4)(2,11)(3,18,21,53)(5,29)(6,27)(7,24)(8,25)(9,28)(10,35,37,63)(12,42)(13,43)(14,22)(15,46)(16,41)(17,48,47,50)(19,54,55,60)(20,56,34,52)(23,33,51,62)(26,31)(30,61)(32,40)(36,64,38,59)(39,45,57,49)(44,58);(1,5,8,30,9,31,6,7)(2,12,16,15,14,44,13,32)(3,19,34,62,21,55,20,33)(4,24,27,26,28,61,25,29)(10,36,47,45,37,38,17,49)(11,40,43,58,22,46,41,42)(18,51,56,60,53,23,52,54)(35,39,48,59,63,57,50,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=eh,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=de,e^c=ei,g^c=ghi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415758);