Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #387 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13846 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
755 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)(3,15,19,32)(4,20)(5,11)(6,12)(7,13)(8,14)(9,18,33,28)(10,23)(16,30,45,48)(17,42,47,56)(21,34,44,41)(22,49)(24,36)(25,38)(26,39)(27,40)(29,37)(31,46,57,53)(35,51)(43,55,63,64)(50,58,62,61)(52,59)(54,60);(1,3,6,18,8,19,7,28)(2,9,13,32,14,33,12,15)(4,21,23,16,24,44,29,45)(5,17,25,46,27,47,26,53)(10,34,20,48,37,41,36,30)(11,31,39,56,40,57,38,42)(22,50,51,43,52,62,54,63)(35,58,49,64,60,61,59,55);(1,4,6,23,8,24,7,29)(2,10,13,20,14,37,12,36)(3,16,18,44,19,45,28,21)(5,22,25,51,27,52,26,54)(9,30,32,34,33,48,15,41)(11,35,39,49,40,60,38,59)(17,43,46,62,47,63,53,50)(31,55,56,58,57,64,42,61);(1,5,6,25,8,27,7,26)(2,11,12,38,14,40,13,39)(3,17,18,46,19,47,28,53)(4,22,23,51,24,52,29,54)(9,31,15,42,33,57,32,56)(10,35,36,59,37,60,20,49)(16,43,44,62,45,63,21,50)(30,55,41,61,48,64,34,58)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265839);