Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14718 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2096 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)(3,14)(4,20)(5,11)(6,29)(7,12)(8,13)(9,17)(10,23)(15,30)(16,40)(18,33)(19,32)(21,34)(22,50)(24,36)(25,48)(26,38)(27,37)(28,39)(31,44)(35,52)(41,61)(42,55)(43,56)(45,57)(46,58)(47,53)(49,64)(51,60)(54,63)(59,62);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,31)(12,32)(13,33)(14,29)(15,24)(20,30)(22,49)(23,43)(25,42)(26,45)(27,44)(28,46)(35,59)(36,55)(37,58)(38,40)(39,57)(41,52)(47,54)(48,56)(50,61)(51,63)(53,62)(60,64);(1,4,6,23,8,25,7,24)(2,10,12,20,13,36,29,48)(3,15,17,42,19,43,18,21)(5,22,26,51,28,53,27,52)(9,30,14,34,33,56,32,55)(11,35,37,47,39,60,38,50)(16,41,44,62,46,63,45,49)(31,54,40,59,57,61,58,64);(1,5,8,28)(2,11,13,39)(3,16,19,46)(4,22,25,53)(6,26,7,27)(9,31,33,57)(10,35,36,60)(12,37,29,38)(14,40,32,58)(15,41,43,63)(17,44,18,45)(20,47,48,50)(21,49,42,62)(23,51,24,52)(30,54,56,61)(34,59,55,64)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,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, 301827);