Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500489 |
| 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 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(9,10,15,20)(11,21,26,44)(12,27,30,17)(13,18,31,40)(14,33,22,19)(16,34,37,25)(23,47,50,42)(24,43,51,65)(28,54,56,38)(29,39,45,55)(32,58,46,41)(35,49,53,59)(36,60,62,52)(48,67,68,64)(57,63,66,71)(61,69,70,72);(1,2)(3,6)(4,5)(7,8)(9,11,22,34,15,26,14,25)(10,16,33,44,20,37,19,21)(12,23,45,59,30,50,29,49)(13,24,46,60,31,51,32,52)(17,35,55,47,27,53,39,42)(18,36,58,65,40,62,41,43)(28,48,66,72,56,68,57,69)(38,61,71,67,54,70,63,64);(2,5)(3,7)(9,12,14,29,15,30,22,45)(10,17,19,39,20,27,33,55)(11,23,25,49,26,50,34,59)(13,28,32,57,31,56,46,66)(16,35,21,42,37,53,44,47)(18,38,41,63,40,54,58,71)(24,48,52,69,51,68,60,72)(36,61,43,64,62,70,65,67);(1,3,4,7)(2,6,5,8)(9,13,15,31)(10,18,20,40)(11,24,26,51)(12,28,30,56)(14,32,22,46)(16,36,37,62)(17,38,27,54)(19,41,33,58)(21,43,44,65)(23,48,50,68)(25,52,34,60)(29,57,45,66)(35,61,53,70)(39,63,55,71)(42,64,47,67)(49,69,59,72);(2,5)(6,8)(13,31)(18,40)(24,51)(28,56)(32,46)(36,62)(38,54)(41,58)(43,65)(48,68)(52,60)(57,66)(61,70)(63,71)(64,67)(69,72)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421590);