Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7548 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1424 |
| commutator | C32 | C42 |
| Frattini, cosocle | C22×C32 | C22 |
| prime radical | C23 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2)(3,4)(6,13,14,33)(7,19)(8,22)(9,26)(10,27)(11,30)(15,41,37,64)(16,43,38,61)(17,47,39,67)(18,48,40,36)(20,29)(21,54)(23,51)(24,56)(25,55)(28,53)(31,59)(32,60,45,34)(35,63,62,44)(42,66,65,46)(49,50)(52,68)(57,58);(2,4)(5,6,8,15,9,16,21,35,27,44,50,43,58,64,7,13,11,18,19,34,28,46,49,47,31,39,54,65,20,32,22,36,12,14,25,37,29,38,56,62,59,63,52,61,53,41,23,33,30,40,51,60,57,66,68,67,10,17,24,42,26,45,55,48)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,e^2=f,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72286491, #72286831, #72286849, … |
| maximal normal subgroup of | #72286491, #72286831, #72286849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1960);