Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405774 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14918 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,27)(3,14,17,33,18,9,30,41)(4,19,22,10,23,37,28,35)(5,24,29,38,25,11,26,36)(15,42,47,31,20,49,44,39)(16,32,45,58,46,40,48,57)(21,50,52,55,53,59,54,34)(43,61,63,60,51,64,62,56);(1,3)(2,9,13,14)(4,20,23,15)(5,16)(6,18)(7,17)(8,30)(10,31)(11,32,24,40)(12,33,27,41)(19,49)(21,51,53,43)(22,47,28,44)(25,46)(26,45)(29,48)(34,56)(35,39)(36,57,38,58)(37,42)(50,64)(52,63,54,62)(55,60)(59,61);(1,4)(2,10)(3,15)(5,21)(6,23)(7,28)(8,22)(9,31)(11,34)(12,37)(13,35)(14,39)(16,43)(17,47)(18,20)(19,27)(24,55)(25,53)(26,54)(29,52)(30,44)(32,56)(33,42)(36,50)(38,59)(40,60)(41,49)(45,63)(46,51)(48,62)(57,64)(58,61);(1,5,6,25)(2,11,13,24)(3,16,18,46)(4,21,23,53)(7,29,8,26)(9,32,14,40)(10,34,35,55)(12,38,27,36)(15,43,20,51)(17,48,30,45)(19,50,37,59)(22,54,28,52)(31,56,39,60)(33,58,41,57)(42,61,49,64)(44,63,47,62)Polycyclic presentation:
2^9|a^2=ef,b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312971);