Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,16,21,15,37)(11,22,25,41,26,40,36,17)(12,27,30,43,29,54,31,18)(13,19,33,45,35,46,34,62)(23,38,50,47,49,64,59,52)(24,44,32,60,51,65,61,39)(28,53,57,67,56,70,58,42)(48,63,55,66,68,72,71,69);(1,2)(3,7)(4,6)(5,8)(9,11,16,26)(10,17,21,41)(12,23,29,49)(13,32,35,61)(14,36,15,25)(18,38,43,64)(19,44,46,65)(20,40,37,22)(24,34,51,33)(27,52,54,47)(28,55,56,71)(30,59,31,50)(39,62,60,45)(42,66,67,69)(48,58,68,57)(53,63,70,72);(1,3)(2,5)(4,8)(6,7)(9,12,15,31,16,29,14,30)(10,18,20,27,21,43,37,54)(11,23,25,50,26,49,36,59)(13,28,34,58,35,56,33,57)(17,38,40,52,41,64,22,47)(19,42,45,53,46,67,62,70)(24,48,32,55,51,68,61,71)(39,63,65,69,60,72,44,66);(3,8)(5,7)(9,13,15,34,16,35,14,33)(10,19,37,62,21,46,20,45)(11,24,36,61,26,51,25,32)(12,28,31,58,29,56,30,57)(17,39,40,65,41,60,22,44)(18,42,54,70,43,67,27,53)(23,48,59,71,49,68,50,55)(38,63,52,69,64,72,47,66);(2,6)(5,7)(12,29)(18,43)(23,49)(27,54)(28,56)(30,31)(38,64)(42,67)(47,52)(48,68)(50,59)(53,70)(55,71)(57,58)(63,72)(66,69)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928581);