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 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,19)(7,21,26,58)(8,27,32,15)(9,31,35,55)(10,20,13,17)(11,38,41,29)(14,45,48,25)(16,44,50,34)(18,37,53,42)(22,33,28,56)(23,61,46,49)(24,64,43,52)(30,40,36,51)(39,54,68,47)(57,62,66,63)(59,65,60,67);(5,7,12,26)(6,14,19,48)(8,28,32,22)(9,33,35,56)(10,24,13,43)(11,39,41,68)(15,23,27,46)(16,49,50,61)(17,47,20,54)(18,52,53,64)(21,29,58,38)(25,37,45,42)(30,60,36,59)(31,66,55,57)(34,67,44,65)(40,62,51,63);(2,4)(5,8,12,32)(6,15,19,27)(7,22,26,28)(9,29,35,38)(10,36,13,30)(11,31,41,55)(14,46,48,23)(16,42,50,37)(17,51,20,40)(18,44,53,34)(21,56,58,33)(24,60,43,59)(25,49,45,61)(39,57,68,66)(47,62,54,63)(52,67,64,65);(5,9)(6,16)(7,23)(8,29)(10,34)(11,40)(12,35)(13,44)(14,22)(15,42)(17,31)(18,36)(19,50)(20,55)(21,57)(24,62)(25,65)(26,46)(27,37)(28,48)(30,53)(32,38)(33,39)(41,51)(43,63)(45,67)(47,59)(49,52)(54,60)(56,68)(58,66)(61,64);(5,10,12,13)(6,17,19,20)(7,24,26,43)(8,30,32,36)(9,34,35,44)(11,37,41,42)(14,47,48,54)(15,40,27,51)(16,31,50,55)(18,29,53,38)(21,52,58,64)(22,59,28,60)(23,62,46,63)(25,39,45,68)(33,67,56,65)(49,66,61,57)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^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,e^c=ei,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, 7530842);