Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1754 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)(3,6)(4,5)(7,8)(9,10,14,19,15,20,35,47)(11,21,24,38,25,46,34,16)(12,26,30,55,29,17,39,42)(13,18,32,44,33,45,62,67)(22,50,52,36,27,48,57,40)(23,49,53,43,31,60,61,37)(28,54,59,71,58,41,64,66)(51,69,70,63,56,68,72,65);(9,11,15,25)(10,16,20,38)(12,27,29,22)(13,31,33,23)(14,34,35,24)(17,40,26,36)(18,43,45,37)(19,46,47,21)(28,56,58,51)(30,52,39,57)(32,53,62,61)(41,65,54,63)(42,48,55,50)(44,49,67,60)(59,70,64,72)(66,68,71,69);(2,5)(6,8)(9,12,14,30,15,29,35,39)(10,17,19,42,20,26,47,55)(11,22,24,52,25,27,34,57)(13,28,32,59,33,58,62,64)(16,36,21,48,38,40,46,50)(18,41,44,66,45,54,67,71)(23,51,53,70,31,56,61,72)(37,63,49,68,43,65,60,69);(9,13,14,32,15,33,35,62)(10,18,19,44,20,45,47,67)(11,23,24,53,25,31,34,61)(12,28,30,59,29,58,39,64)(16,37,21,49,38,43,46,60)(17,41,42,66,26,54,55,71)(22,51,52,70,27,56,57,72)(36,63,48,68,40,65,50,69);(1,3,4,7)(2,6,5,8)(12,29)(17,26)(22,27)(28,58)(30,39)(36,40)(41,54)(42,55)(48,50)(51,56)(52,57)(59,64)(63,65)(66,71)(68,69)(70,72)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363645);