Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #33720 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,6)(7,8)(9,10,14,20)(11,22,24,17)(12,27,29,63)(13,19,16,37)(15,34,33,61)(18,43,45,30)(21,32,47,36)(23,51,52,60)(25,42,31,40)(26,58,56,66)(28,62,38,48)(35,44,65,46)(39,54,49,70)(41,57,50,64)(53,71,55,68)(59,67,72,69);(1,2,5,6)(3,7,4,8)(9,11,14,24)(10,17,20,22)(12,23,29,52)(13,31,16,25)(15,26,33,56)(18,39,45,49)(19,42,37,40)(21,41,47,50)(27,60,63,51)(28,55,38,53)(30,54,43,70)(32,64,36,57)(34,66,61,58)(35,59,65,72)(44,69,46,67)(48,68,62,71);(1,3,5,4)(2,7,6,8)(9,12,14,29)(10,18,20,45)(11,23,24,52)(13,28,16,38)(15,35,33,65)(17,39,22,49)(19,44,37,46)(21,48,47,62)(25,53,31,55)(26,59,56,72)(27,61,63,34)(30,32,43,36)(40,67,42,69)(41,68,50,71)(51,66,60,58)(54,64,70,57);(1,4,5,3)(2,7,6,8)(11,24)(17,22)(23,52)(25,31)(26,56)(39,49)(40,42)(41,50)(51,60)(53,55)(54,70)(57,64)(58,66)(59,72)(67,69)(68,71);(9,13,14,16)(10,19,20,37)(11,25,24,31)(12,28,29,38)(15,32,33,36)(17,40,22,42)(18,44,45,46)(21,34,47,61)(23,53,52,55)(26,57,56,64)(27,62,63,48)(30,35,43,65)(39,67,49,69)(41,58,50,66)(51,71,60,68)(54,72,70,59)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531168);