Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| 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 | |
1935 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)(7,19,23,51)(8,24,28,49)(9,17,11,15)(10,30,34,44)(13,37,40,22)(14,36,43,26)(16,29,47,33)(20,52,56,41)(21,57,35,45)(25,60,62,38)(27,63,53,42)(31,48,64,39)(32,46,65,68)(50,54,59,55)(58,67,61,66);(5,7,12,23)(6,13,18,40)(8,25,28,62)(9,21,11,35)(10,31,34,64)(14,41,43,52)(15,39,17,48)(16,45,47,57)(19,44,51,30)(20,53,56,27)(22,29,37,33)(24,59,49,50)(26,61,36,58)(32,54,65,55)(38,63,60,42)(46,66,68,67);(2,4);(5,8)(6,14)(7,20)(9,26)(10,32)(11,36)(12,28)(13,38)(15,24)(16,46)(17,49)(18,43)(19,50)(21,54)(22,58)(23,56)(25,31)(27,33)(29,53)(30,42)(34,65)(35,55)(37,61)(39,66)(40,60)(41,45)(44,63)(47,68)(48,67)(51,59)(52,57)(62,64);(5,9,12,11)(6,15,18,17)(7,21,23,35)(8,26,28,36)(10,29,34,33)(13,39,40,48)(14,24,43,49)(16,44,47,30)(19,45,51,57)(20,54,56,55)(22,31,37,64)(25,61,62,58)(27,32,53,65)(38,66,60,67)(41,59,52,50)(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,d^b=df,f^b=fh,h^b=hi,f^d=fh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530808);