Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8,11,28)(4,18)(5,23)(9,32,34,26)(10,27,37,56)(12,31,29,19)(14,42)(15,41)(21,51)(22,49)(24,25)(30,35,50,53)(33,59,60,54)(36,57,61,55)(38,58,39,44)(40,43)(45,64)(46,47)(48,52)(62,63);(2,9,11,34)(3,13)(4,18)(5,24)(8,26,28,32)(10,36,37,61)(12,35,29,53)(14,43)(15,46)(16,17)(19,50,31,30)(21,52)(22,49)(23,25)(27,55,56,57)(33,38,60,39)(40,42)(41,47)(44,54,58,59)(48,51);(1,2,7,12)(3,10,17,38)(4,19,18,28)(5,9,25,35)(6,11,20,29)(8,22,31,49)(13,39,16,37)(14,44,42,56)(15,33,46,61)(21,32,48,50)(23,53,24,34)(26,52,30,51)(27,43,58,40)(36,47,60,41)(45,59,63,55)(54,62,57,64);(1,3,6,16)(2,10,11,37)(4,14,22,43)(5,15,24,47)(7,17,20,13)(8,27,28,56)(9,33,34,60)(12,38,29,39)(18,42,49,40)(19,44,31,58)(21,45,51,64)(23,41,25,46)(26,54,32,59)(30,57,50,55)(35,61,53,36)(48,63,52,62)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,d^2=h,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,f^b=fi,g^b=gh,e^c=eh,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145614);