Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #506304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #28165 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
560 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)(4,10)(5,9)(6,27)(7,12)(8,16)(11,24)(13,25)(15,29)(17,32)(18,31)(19,47)(20,33)(21,39)(22,54)(23,28)(26,34)(30,55)(35,57)(36,48)(37,51)(38,52)(40,46)(41,43)(42,45)(44,58)(49,59)(50,63)(53,62)(56,60)(61,64);(1,3)(2,8,12,14)(4,19,21,48)(5,23)(6,18)(7,16)(9,32,34,28)(10,36)(11,40,35,30)(13,31,27,42)(15,22,43,52)(17,26)(20,50,53,49)(24,55)(25,45)(29,54)(33,59)(37,60)(38,41)(39,47)(44,64,56,51)(46,57)(58,61)(62,63);(2,9)(5,24)(6,25)(8,28)(10,33)(11,13)(12,34)(14,32)(17,46)(18,45)(19,48)(20,51)(22,52)(23,55)(26,57)(27,35)(29,58)(30,31)(36,59)(37,38)(39,62)(40,42)(41,60)(44,49)(47,63)(50,56)(53,64)(54,61);(1,4)(2,10)(3,15,16,43)(5,20)(6,22)(7,21)(8,29,14,41)(9,33)(11,37)(12,39)(13,38)(17,44,23,56)(18,19,45,48)(24,51)(25,52)(26,53)(27,54)(28,58,32,60)(30,59,40,63)(31,36,42,47)(34,62)(35,61)(46,49,55,50)(57,64)Polycyclic presentation:
2^9|b^2=g,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fi,f^c=fh,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413500);