Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2097 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)(5,11)(6,25)(7,10)(8,15)(9,20)(12,27)(14,40)(16,46)(17,38)(19,50)(21,36)(22,52)(23,55)(24,33)(26,31)(28,58)(29,45)(30,60)(32,51)(34,54)(35,56)(37,42)(39,48)(41,49)(43,63)(44,57)(47,62)(53,61)(59,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,28)(10,29)(11,13)(12,30)(18,49)(19,41)(20,42)(21,43)(22,53)(23,44)(24,45)(25,57)(26,47)(27,48)(31,59)(32,37)(33,38)(34,62)(35,39)(36,61)(40,50)(46,55)(51,58)(52,63)(54,64)(56,60);(1,4,7,21)(2,9,10,31)(3,14,17,43)(5,22,24,19)(6,26,27,20)(8,28,29,59)(11,34,33,32)(12,36,25,18)(13,37,38,62)(15,41,45,53)(16,47,48,42)(23,51,56,54)(30,61,57,49)(35,50,55,52)(39,63,46,40)(44,64,60,58);(2,10)(8,29)(9,31)(11,33)(12,25)(13,38)(18,36)(28,59)(30,57)(32,34)(35,55)(37,62)(39,46)(40,63)(49,61)(50,52)Polycyclic presentation:
2^9|c^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,h^b=hi,e^c=eh,f^c=fi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72280488, #72281917, #72281946, … |
| maximal normal subgroup of | #72280488, #72281917, #72281946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406095);