Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499825 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| 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 | |
1831 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)(5,10)(6,25)(7,12)(8,15)(11,28)(14,44)(16,42)(17,50)(18,52)(19,36)(20,34)(21,40)(22,33)(23,37)(24,58)(26,43)(27,35)(29,47)(30,59)(31,49)(32,53)(38,57)(39,48)(41,45)(46,60)(51,62)(54,63)(55,61)(56,64);(1,3,7,16)(2,8,12,31)(4,18,19,51)(5,15,23,49)(6,26,28,17)(9,32,36,61)(10,13,37,42)(11,39,25,30)(14,27,45,21)(20,53,22,55)(24,59,57,48)(29,40,60,35)(33,62,34,52)(38,50,58,43)(41,63,44,64)(46,56,47,54);(1,4)(2,9)(3,14)(5,22)(6,27)(7,19)(8,29)(10,34)(11,40)(12,36)(13,41)(15,47)(16,45)(17,51)(18,26)(20,23)(21,28)(24,54)(25,35)(30,61)(31,60)(32,39)(33,37)(38,63)(42,44)(43,62)(46,49)(48,53)(50,52)(55,59)(56,57)(58,64);(2,10)(4,19)(8,13)(9,33)(11,38)(12,37)(14,45)(18,51)(20,22)(21,27)(25,58)(29,44)(30,43)(31,42)(32,62)(34,36)(35,63)(39,50)(40,64)(41,60)(46,47)(52,61)(53,55)(54,56)Polycyclic presentation:
2^9|b^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72281882, #72281917, #72281946, … |
| maximal normal subgroup of | #72281882, #72281917, #72281946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407022);