Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,18,34,17)(7,19,22,37,24,49,23,56)(8,14,27,42,28,44,59,43)(9,15,30,45,31,46,10,32)(13,35,38,55,39,33,47,21)(20,48,52,65,54,68,53,62)(25,40,26,57,29,41,60,66)(36,61,63,50,64,58,67,51);(1,2,4,3)(5,7)(6,13)(8,20)(9,21)(10,33)(11,22)(12,24)(14,36)(15,37)(16,38)(17,47)(18,39)(19,32)(23,34)(25,50)(26,58)(27,52)(28,54)(29,51)(30,35)(31,55)(40,62)(41,65)(42,63)(43,67)(44,64)(45,49)(46,56)(48,57)(53,59)(60,61)(66,68);(1,3,4,2)(5,8)(6,14)(7,20)(9,29)(10,26)(11,27)(12,28)(13,36)(15,41)(16,42)(17,43)(18,44)(19,48)(21,51)(22,52)(23,53)(24,54)(25,31)(30,60)(32,57)(33,58)(34,59)(35,61)(37,65)(38,63)(39,64)(40,46)(45,66)(47,67)(49,68)(50,55)(56,62);(5,9,11,30,12,31,34,10)(6,15,16,45,18,46,17,32)(7,21,22,35,24,55,23,33)(8,25,27,26,28,29,59,60)(13,37,38,49,39,56,47,19)(14,40,42,57,44,41,43,66)(20,50,52,58,54,51,53,61)(36,62,63,48,64,65,67,68)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400440);