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 | |
1924 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)(7,17)(8,22)(9,27)(10,33)(11,14)(12,20)(13,25)(15,31)(16,30)(18,48)(19,52)(21,57)(23,60)(24,63)(26,46)(28,45)(29,44)(32,47)(34,43)(35,40)(36,51)(37,61)(38,62)(39,50)(41,53)(42,55)(49,54)(56,66)(58,59)(64,67)(65,68);(5,7,11,18)(6,12,14,36)(8,23,26,61)(9,28,32,62)(10,34,30,21)(13,39,40,52)(15,43,42,57)(16,45,33,38)(17,47,48,27)(19,53,50,24)(20,55,51,31)(22,59,46,49)(25,64,35,56)(29,58,65,54)(37,63,60,41)(44,67,68,66);(2,4)(7,18)(12,36)(17,48)(19,50)(20,51)(21,34)(23,61)(28,62)(37,60)(38,45)(39,52)(43,57)(49,59)(54,58)(56,64)(66,67);(5,8,11,26)(6,13,14,40)(7,19,18,50)(9,29,32,65)(10,35,30,25)(12,37,36,60)(15,44,42,68)(16,46,33,22)(17,49,48,59)(20,56,51,64)(21,58,34,54)(23,62,61,28)(24,55,53,31)(27,41,47,63)(38,67,45,66)(39,57,52,43);(6,14)(12,36)(13,40)(15,42)(16,33)(17,48)(22,46)(27,47)(37,60)(38,45)(39,52)(41,63)(43,57)(44,68)(49,59)(66,67)Polycyclic presentation:
2^9|b^2=i,d^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, 7530812);