Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1930 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)(5,6,11,15)(7,13,19,43)(8,22,26,50)(9,27,32,64)(10,33,30,17)(12,36,39,20)(14,35,45,25)(16,48,47,31)(18,51,52,44)(21,55,34,42)(23,58,60,37)(24,61,62,46)(28,41,59,38)(29,49,65,68)(40,56,57,53)(54,66,63,67);(5,7)(6,12)(8,23)(9,28)(10,34)(11,19)(13,27)(14,44)(15,39)(16,42)(17,41)(18,24)(20,31)(21,30)(22,57)(25,63)(26,60)(29,56)(32,59)(33,38)(35,54)(36,48)(37,46)(40,50)(43,64)(45,51)(47,55)(49,67)(52,62)(53,65)(58,61)(66,68);(2,4)(5,7)(6,13)(8,18)(9,20)(10,21)(11,19)(12,27)(14,40)(15,43)(16,41)(17,42)(22,51)(23,24)(25,53)(26,52)(28,31)(29,54)(30,34)(32,36)(33,55)(35,56)(37,46)(38,47)(39,64)(44,50)(45,57)(48,59)(49,66)(58,61)(60,62)(63,65)(67,68);(5,8,11,26)(6,14,15,45)(7,18,19,52)(9,29,32,65)(10,35,30,25)(12,37,39,58)(13,40,43,57)(16,49,47,68)(17,50,33,22)(20,54,36,63)(21,56,34,53)(23,59,60,28)(24,48,62,31)(27,46,64,61)(38,67,41,66)(42,44,55,51);(6,15)(7,19)(14,45)(16,47)(17,33)(18,52)(20,36)(21,34)(22,50)(23,60)(27,64)(28,59)(46,61)(49,68)(53,56)(54,63)Polycyclic presentation:
2^9|a^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,d^b=df,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530847);