Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479749 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17233 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
731 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)(3,15,19,43)(4,20,24,10)(5,26,30,41)(6,12)(7,14)(8,13)(9,18,35,32)(11,31,42,29)(16,44)(17,34)(21,53)(22,40)(23,38,25,37)(27,45)(28,52)(33,55)(36,56)(39,49)(46,59,54,62)(47,51)(48,60,64,63)(50,58)(57,61);(1,3,7,18,8,19,6,32)(2,9,14,15,13,35,12,43)(4,21,25,55,24,16,23,47)(5,27,31,17,30,49,29,50)(10,33,38,44,20,51,37,53)(11,39,26,58,42,45,41,34)(22,48,56,54,57,64,28,46)(36,60,61,62,52,63,40,59);(1,4,6,23,8,24,7,25)(2,10,12,37,13,20,14,38)(3,16,18,47,19,21,32,55)(5,28,29,57,30,56,31,22)(9,33,15,44,35,51,43,53)(11,40,41,52,42,61,26,36)(17,48,49,54,50,64,27,46)(34,60,39,62,58,63,45,59);(1,5,8,30)(2,11,13,42)(3,17,19,50)(4,22,24,57)(6,31,7,29)(9,34,35,58)(10,36,20,52)(12,26,14,41)(15,45,43,39)(16,46,21,54)(18,27,32,49)(23,56,25,28)(33,59,51,62)(37,61,38,40)(44,63,53,60)(47,64,55,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,g^c=gi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386946);