Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14727 |
| 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 | |
1728 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)(3,15,19,37)(4,10)(5,24,26,39)(6,12)(7,14)(8,13)(9,18,32,27)(11,25,36,29)(16,42,20,40)(17,31)(21,50,52,55)(22,34)(23,35)(28,38)(30,45,57,44)(33,53,59,51)(41,46)(43,56)(47,54)(48,58)(49,64)(60,62)(61,63);(1,3)(2,9)(4,20)(5,17)(6,27)(7,18)(8,19)(10,30)(11,31)(12,37)(13,32)(14,15)(16,23)(21,49)(22,44)(24,54)(25,47)(26,48)(28,45)(29,46)(33,56)(34,42)(35,57)(36,58)(38,40)(39,41)(43,52)(50,60)(51,62)(53,63)(55,61)(59,64);(1,4,6,22,8,23,7,28)(2,10,12,34,13,35,14,38)(3,16,18,44,19,20,27,45)(5,21,25,51,26,52,29,53)(9,30,15,40,32,57,37,42)(11,33,24,55,36,59,39,50)(17,43,46,62,48,49,47,63)(31,56,41,60,58,64,54,61);(1,5,8,26)(2,11,13,36)(3,17,19,48)(4,21,23,52)(6,25,7,29)(9,31,32,58)(10,33,35,59)(12,24,14,39)(15,41,37,54)(16,43,20,49)(18,46,27,47)(22,51,28,53)(30,56,57,64)(34,55,38,50)(40,60,42,61)(44,62,45,63)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303016);