Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,9,14,10,15,24,34,11,16,25,35,26,36,46,56)(7,12,19,29,20,30,42,52,18,28,40,50,41,51,60,65)(8,13,22,32,23,33,45,55,21,31,43,53,44,54,61,66)(17,27,38,48,39,49,59,64,37,47,57,62,58,63,67,68);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,27)(14,29)(15,30)(16,28)(21,37)(22,38)(23,39)(24,42)(25,40)(26,41)(31,47)(32,48)(33,49)(34,52)(35,50)(36,51)(43,57)(44,58)(45,59)(46,60)(53,62)(54,63)(55,64)(56,65)(61,67)(66,68);(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,27)(14,32)(15,33)(16,31)(18,37)(19,38)(20,39)(24,45)(25,43)(26,44)(28,47)(29,48)(30,49)(34,55)(35,53)(36,54)(40,57)(41,58)(42,59)(46,61)(50,62)(51,63)(52,64)(56,66)(60,67)(65,68);(8,21)(13,31)(17,37)(22,43)(23,44)(27,47)(32,53)(33,54)(38,57)(39,58)(45,61)(48,62)(49,63)(55,66)(59,67)(64,68);(7,18)(12,28)(17,37)(19,40)(20,41)(27,47)(29,50)(30,51)(38,57)(39,58)(42,60)(48,62)(49,63)(52,65)(59,67)(64,68)Polycyclic presentation:
2^9|a^2=g,b^2=f,g^2=h,h^2=i,b^a=bf,e^b=ei,d^c=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529716);