Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478920 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17175 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
773 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)(4,10,24,38)(5,11)(6,13)(7,14)(8,12)(9,18)(16,45,47,35)(17,48)(19,33)(20,34)(21,53,52,31)(22,55,56,36)(23,44,30,37)(25,32)(26,40,57,64)(27,42)(28,43)(29,41)(39,51)(46,60,58,62)(49,59,54,63)(50,61);(1,3,8,18,7,20,6,19)(2,9,12,34,14,33,13,15)(4,21,30,47,24,52,23,16)(5,25,29,51,28,50,27,17)(10,35,37,31,38,45,44,53)(11,39,41,61,43,48,42,32)(22,46,57,49,56,58,26,54)(36,63,40,60,55,59,64,62);(1,4,8,30,7,24,6,23)(2,10,12,37,14,38,13,44)(3,16,18,21,20,47,19,52)(5,26,29,22,28,57,27,56)(9,31,34,45,33,53,15,35)(11,40,41,55,43,64,42,36)(17,49,25,58,51,54,50,46)(32,60,39,59,61,62,48,63);(1,5,7,28)(2,11,14,43)(3,17,20,51)(4,22,24,56)(6,29,8,27)(9,32,33,61)(10,36,38,55)(12,42,13,41)(15,39,34,48)(16,46,47,58)(18,50,19,25)(21,54,52,49)(23,57,30,26)(31,59,53,63)(35,62,45,60)(37,64,44,40)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386117);