Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #463937 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16497 |
| 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 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,7,14,6,12)(3,15,18,35,20,34,19,9)(4,21,24,39,26,38,25,10)(5,27,31,41,30,11,29,42)(16,32,22,43,47,53,46,36)(17,48,50,33,52,44,51,58)(23,55,28,37,57,40,56,61)(45,62,54,60,64,59,49,63);(1,3,6,18,7,20,8,19)(2,9,13,34,14,35,12,15)(4,22,25,47,26,46,24,16)(5,17,29,50,30,52,31,51)(10,36,21,53,39,43,38,32)(11,33,42,48,27,58,41,44)(23,54,56,64,57,49,28,45)(37,60,40,62,61,63,55,59);(1,4,6,25,7,26,8,24)(2,10,13,21,14,39,12,38)(3,16,18,22,20,47,19,46)(5,28,29,23,30,56,31,57)(9,32,34,36,35,53,15,43)(11,40,42,61,27,55,41,37)(17,49,50,45,52,54,51,64)(33,60,48,62,58,63,44,59);(1,5,7,30)(2,11)(3,17,20,52)(4,23)(6,29,8,31)(9,33)(10,37,39,61)(12,41)(13,42)(14,27)(15,44)(16,45)(18,50,19,51)(21,40,38,55)(22,54)(24,28)(25,56)(26,57)(32,59,53,62)(34,48)(35,58)(36,60,43,63)(46,49)(47,64)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371133);