Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501196 |
| 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 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1054 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)(8,15,30,47)(14,41)(16,42)(17,43)(18,50)(19,36)(20,33,23,34)(21,35)(25,53,56,37)(26,58)(27,39)(28,40)(29,45,60,46)(31,48,38,57)(32,51,61,54)(44,59)(49,62)(52,63,55,64);(1,3,7,17)(2,8,12,30)(4,18,19,49)(5,15,24,47)(6,26,28,16)(9,32,36,61)(10,13,22,43)(11,38,40,31)(14,27,44,21)(20,51,23,54)(25,57,56,48)(29,39,60,35)(33,62,34,50)(37,58,53,42)(41,63,59,64)(45,55,46,52);(1,4)(2,9)(3,14)(5,23)(6,27)(7,19)(8,29)(10,34)(11,39)(12,36)(13,41)(15,46)(16,49)(17,44)(18,26)(20,24)(21,28)(22,33)(25,52)(30,60)(31,61)(32,38)(35,40)(37,63)(42,62)(43,59)(45,47)(48,51)(50,58)(53,64)(54,57)(55,56);(2,10)(4,19)(8,13)(9,33)(11,37)(12,22)(14,44)(18,49)(20,23)(21,27)(29,59)(30,43)(31,42)(32,62)(34,36)(35,63)(38,58)(39,64)(40,53)(41,60)(45,46)(50,61)(51,54)(52,55)Polycyclic presentation:
2^9|a^2=f,b^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,d^c=di,e^c=ehi,g^c=gi,f^d=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72303720, #72303743, #72303747, … |
| maximal normal subgroup of | #72303720, #72303743, #72303747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408396);