Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16873 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,18,34,19,33,20,9)(4,10,23,37,24,39,25,38)(5,26,28,42,29,41,30,11)(16,35,21,52,47,43,46,31)(17,32,49,60,50,61,51,44)(22,40,27,56,55,63,54,36)(45,58,48,59,57,64,53,62);(1,3,7,18,8,19,6,20)(2,9,14,15,13,34,12,33)(4,21,25,47,24,46,23,16)(5,17,28,51,29,50,30,49)(10,35,38,52,39,43,37,31)(11,32,26,44,42,61,41,60)(22,53,27,57,55,48,54,45)(36,62,40,64,56,59,63,58);(1,4,7,23,8,24,6,25)(2,10,12,38,13,39,14,37)(3,16,18,46,19,47,20,21)(5,27,28,22,29,54,30,55)(9,31,33,35,34,52,15,43)(11,40,41,56,42,63,26,36)(17,48,51,57,50,53,49,45)(32,59,60,58,61,62,44,64);(1,5,6,30,8,29,7,28)(2,11,14,26,13,42,12,41)(3,17,20,49,19,50,18,51)(4,22,25,27,24,55,23,54)(9,32,15,44,34,61,33,60)(10,36,37,63,39,56,38,40)(16,45,21,53,47,57,46,48)(31,58,43,59,52,64,35,62)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=efi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380794);