Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478231 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17136 |
| 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 | |
829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,28)(3,9,16,32,17,34,29,33)(4,18,22,37,21,41,23,10)(5,11,26,40,25,39,27,57)(14,42,19,50,44,35,48,30)(15,45,47,38,46,31,24,56)(20,49,53,60,55,62,54,36)(43,59,51,58,52,61,64,63);(1,3,6,16,8,17,7,29)(2,9,12,32,13,34,28,33)(4,19,23,14,21,48,22,44)(5,24,27,46,25,47,26,15)(10,35,41,50,37,42,18,30)(11,38,57,45,39,56,40,31)(20,52,53,64,55,43,54,51)(36,58,49,61,60,63,62,59);(1,4,6,23,8,21,7,22)(2,10,12,41,13,37,28,18)(3,14,16,48,17,44,29,19)(5,20,27,53,25,55,26,54)(9,30,32,35,34,50,33,42)(11,36,57,49,39,60,40,62)(15,43,24,51,46,52,47,64)(31,58,38,61,45,63,56,59);(1,5,7,27,8,25,6,26)(2,11,28,57,13,39,12,40)(3,15,29,24,17,46,16,47)(4,20,23,54,21,55,22,53)(9,31,33,38,34,45,32,56)(10,36,41,62,37,60,18,49)(14,43,48,64,44,52,19,51)(30,58,35,59,50,63,42,61)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ef,d^2=fi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385428);