Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471017 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16731 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
608 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)(3,13)(4,18)(5,10,23,34)(6,25)(7,12)(8,16)(9,20)(11,27)(14,40)(15,29,45,38)(17,30)(19,49,22,51)(21,36)(24,55,54,35)(26,32)(28,47)(31,53,33,50)(37,43)(39,46)(41,61,52,60)(42,48)(44,59,64,57)(56,63,58,62);(1,3)(2,8,12,13)(4,14)(5,15)(6,17)(7,16)(9,28,32,37)(10,29,34,38)(11,30,25,39)(18,48,36,40)(19,41)(20,43)(21,42)(22,52)(23,45)(24,44)(26,47)(27,46)(31,56,33,58)(35,57,55,59)(49,61,51,60)(50,62)(53,63)(54,64);(1,4,7,21)(2,9,12,32)(3,14,16,42)(5,22,23,19)(6,26,27,20)(8,28,13,37)(10,33,34,31)(11,36,25,18)(15,41,45,52)(17,47,46,43)(24,50,54,53)(29,56,38,58)(30,40,39,48)(35,49,55,51)(44,63,64,62)(57,60,59,61);(1,5,6,24,7,23,27,54)(2,10,11,35,12,34,25,55)(3,15,17,44,16,45,46,64)(4,19,20,50,21,22,26,53)(8,29,30,57,13,38,39,59)(9,31,18,49,32,33,36,51)(14,41,43,62,42,52,47,63)(28,56,48,61,37,58,40,60)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=i,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378214);