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 ◁ #532 ◁ #2931 ◁ 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)(5,6,12,19)(7,20,25,53)(8,14,28,44)(9,18,11,16)(10,31,35,47)(13,37,41,24)(15,29,42,26)(17,30,49,34)(21,51,57,39)(22,58,61,45)(23,62,36,48)(27,63,59,46)(32,50,65,40)(33,43,64,68)(38,60,52,55)(54,66,56,67);(5,7,12,25)(6,13,19,41)(8,21,28,57)(9,23,11,36)(10,32,35,65)(14,38,44,52)(15,45,42,58)(16,40,18,50)(17,48,49,62)(20,47,53,31)(22,59,61,27)(24,30,37,34)(26,54,29,56)(33,55,64,60)(39,63,51,46)(43,66,68,67);(2,4)(5,8)(6,14)(7,21)(9,29)(10,27)(11,26)(12,28)(13,38)(15,16)(17,43)(18,42)(19,44)(20,51)(22,32)(23,56)(24,55)(25,57)(30,64)(31,63)(33,34)(35,59)(36,54)(37,60)(39,53)(40,45)(41,52)(46,47)(48,66)(49,68)(50,58)(61,65)(62,67);(5,8)(6,15)(7,22)(9,26)(10,33)(11,29)(12,28)(13,39)(14,16)(17,43)(18,44)(19,42)(20,52)(21,32)(23,55)(24,56)(25,61)(27,34)(30,59)(31,46)(35,64)(36,60)(37,54)(38,53)(40,66)(41,51)(45,48)(47,63)(49,68)(50,67)(57,65)(58,62);(5,9,12,11)(6,16,19,18)(7,23,25,36)(8,26,28,29)(10,30,35,34)(13,40,41,50)(14,42,44,15)(17,47,49,31)(20,48,53,62)(21,54,57,56)(22,55,61,60)(24,32,37,65)(27,33,59,64)(38,58,52,45)(39,66,51,67)(43,63,68,46)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,e^c=ei,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, 7530834);