Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2331 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14266 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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:
(1,2,7,13,8,14,6,12)(3,15,18,33,19,32,29,9)(4,20,22,36,24,41,23,10)(5,11,27,39,28,40,26,38)(16,30,46,42,47,34,45,56)(17,43,49,59,50,58,54,31)(21,37,25,53,52,62,51,35)(44,55,63,61,64,60,48,57);(1,3,6,18,8,19,7,29)(2,9,13,32,14,33,12,15)(4,16,23,46,24,47,22,45)(5,17,26,49,28,50,27,54)(10,34,20,42,36,30,41,56)(11,31,39,58,40,59,38,43)(21,44,51,63,52,64,25,48)(35,60,37,61,53,55,62,57);(1,4,6,23,8,24,7,22)(2,10,13,20,14,36,12,41)(3,16,18,46,19,47,29,45)(5,25,26,21,28,51,27,52)(9,30,32,56,33,34,15,42)(11,37,39,53,40,62,38,35)(17,48,49,44,50,63,54,64)(31,57,58,60,59,61,43,55);(1,5,6,26,8,28,7,27)(2,11,12,38,14,40,13,39)(3,17,18,49,19,50,29,54)(4,21,22,25,24,52,23,51)(9,31,15,43,33,59,32,58)(10,35,20,37,36,53,41,62)(16,44,45,48,47,64,46,63)(30,55,56,57,34,60,42,61)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287588);