Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2044 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1075 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)(4,20)(5,26)(6,28)(7,12)(8,11)(9,16)(10,23)(13,30)(15,43)(17,48)(18,33)(19,38)(21,53)(22,52)(24,51)(25,50)(27,39)(29,42)(31,41)(32,60)(34,49)(35,45)(36,56)(37,57)(40,47)(44,62)(46,64)(54,61)(55,58)(59,63);(1,3,8,19)(2,9,11,33)(4,21,25,44)(5,18,7,16)(6,29,30,17)(10,35,36,63)(12,38,26,14)(13,40,28,34)(15,22,46,24)(20,43,50,64)(23,54,56,60)(27,47,31,49)(32,58,61,37)(39,42,41,48)(45,55,59,57)(51,62,52,53);(1,4,5,22,8,25,7,24)(2,10,12,37,11,36,26,58)(3,15,16,44,19,46,18,21)(6,23,27,55,30,56,31,57)(9,32,14,35,33,61,38,63)(13,20,39,51,28,50,41,52)(17,45,47,60,29,59,49,54)(34,62,42,43,40,53,48,64);(2,11)(9,33)(10,36)(12,26)(13,28)(14,38)(20,50)(32,61)(34,40)(35,63)(37,58)(39,41)(42,48)(43,64)(51,52)(53,62)Polycyclic presentation:
2^9|b^2=i,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72303725, #72303728, #72303741, … |
| maximal normal subgroup of | #72303725, #72303728, #72303741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406323);