Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501188 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #23885 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1048 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)(4,9)(5,22,24,10)(6,11)(7,12)(8,15,30,46)(14,40)(16,41)(17,42)(18,49)(19,51,23,33)(20,34)(21,35)(25,53,56,36)(26,58)(27,38)(28,39)(29,43,59,45)(31,47,37,57)(32,50,60,54)(44,63)(48,64)(52,62,55,61);(1,3,7,17)(2,8,12,30)(4,18,21,48)(5,15,24,46)(6,26,28,16)(9,32,35,60)(10,13,22,42)(11,37,39,31)(14,27,44,20)(19,50,23,54)(25,57,56,47)(29,38,59,34)(33,49,51,64)(36,58,53,41)(40,62,63,61)(43,55,45,52);(1,4,7,21)(2,9,12,35)(3,14,17,44)(5,23,24,19)(6,27,28,20)(8,29,30,59)(10,33,22,51)(11,38,39,34)(13,40,42,63)(15,45,46,43)(16,48,26,18)(25,52,56,55)(31,60,37,32)(36,62,53,61)(41,64,58,49)(47,50,57,54);(2,10)(8,13)(9,33)(11,36)(12,22)(29,40)(30,42)(31,41)(32,49)(34,61)(35,51)(37,58)(38,62)(39,53)(59,63)(60,64)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=ehi,g^c=gi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72303731, #72303743, #72303746, … |
| maximal normal subgroup of | #72303731, #72303743, #72303746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408390);