Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2337 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14473 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1129 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)(3,15,20,9,19,34,18,33)(4,10,24,38,25,39,23,37)(5,26,28,42,29,41,30,11)(16,35,46,56,45,43,21,31)(17,32,27,40,47,57,48,58)(22,50,52,61,53,60,54,36)(44,63,49,59,51,55,64,62);(1,3)(2,9)(4,21)(5,27)(6,20)(7,18)(8,19)(10,35)(11,40)(12,34)(13,15)(14,33)(16,23)(17,28)(22,51)(24,45)(25,46)(26,32)(29,48)(30,47)(31,38)(36,55)(37,56)(39,43)(41,57)(42,58)(44,53)(49,52)(50,59)(54,64)(60,62)(61,63);(1,4,6,23,8,25,7,24)(2,10,12,37,14,39,13,38)(3,16,18,21,19,45,20,46)(5,22,28,52,29,53,30,54)(9,31,15,43,33,56,34,35)(11,36,26,50,42,61,41,60)(17,44,27,49,47,51,48,64)(32,55,40,62,57,63,58,59);(1,5,6,28,8,29,7,30)(2,11,13,41,14,42,12,26)(3,17,18,27,19,47,20,48)(4,22,23,52,25,53,24,54)(9,32,34,58,33,57,15,40)(10,36,38,60,39,61,37,50)(16,44,21,49,45,51,46,64)(31,55,35,59,56,63,43,62)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294609);