Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,18)(7,19,20,50)(8,24,28,49)(9,17,11,15)(10,30,34,44)(13,37,38,23)(14,36,43,26)(16,29,47,33)(21,53,52,41)(22,57,35,45)(25,60,61,39)(27,63,54,42)(31,48,64,40)(32,46,65,68)(51,55,59,56)(58,67,62,66);(5,7,12,20)(6,13,18,38)(8,25,28,61)(9,22,11,35)(10,31,34,64)(14,41,43,53)(15,40,17,48)(16,45,47,57)(19,44,50,30)(21,54,52,27)(23,29,37,33)(24,59,49,51)(26,62,36,58)(32,55,65,56)(39,63,60,42)(46,66,68,67);(2,4)(7,20)(13,38)(19,50)(21,52)(22,35)(23,37)(25,61)(31,64)(39,60)(40,48)(41,53)(45,57)(51,59)(55,56)(58,62)(66,67);(5,8)(6,14)(7,21)(9,26)(10,32)(11,36)(12,28)(13,39)(15,24)(16,46)(17,49)(18,43)(19,51)(20,52)(22,55)(23,58)(25,31)(27,33)(29,54)(30,42)(34,65)(35,56)(37,62)(38,60)(40,66)(41,45)(44,63)(47,68)(48,67)(50,59)(53,57)(61,64);(5,9,12,11)(6,15,18,17)(7,22,20,35)(8,26,28,36)(10,29,34,33)(13,40,38,48)(14,24,43,49)(16,44,47,30)(19,45,50,57)(21,55,52,56)(23,31,37,64)(25,62,61,58)(27,32,54,65)(39,66,60,67)(41,59,53,51)(42,46,63,68)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fh,h^b=hi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530815);