Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629034 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2104 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)(9,10,15,20)(11,21,25,44)(12,17,28,39)(13,30,32,18)(14,33,22,19)(16,34,37,26)(23,42,49,65)(24,50,51,43)(27,53,54,38)(29,56,45,40)(31,41,46,58)(35,59,62,48)(36,52,57,60)(47,67,69,64)(55,63,66,71)(61,68,70,72);(1,3,4,7)(2,5,6,8)(9,11,22,34,15,25,14,26)(10,16,33,44,20,37,19,21)(12,23,45,59,28,49,29,48)(13,24,46,60,32,51,31,52)(17,35,56,65,39,62,40,42)(18,36,58,50,30,57,41,43)(27,47,66,72,54,69,55,68)(38,61,71,67,53,70,63,64);(2,6)(5,8)(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,35)(18,38)(19,40)(20,39)(21,42)(22,45)(24,47)(25,49)(26,48)(30,53)(31,55)(32,54)(33,56)(34,59)(36,61)(37,62)(41,63)(43,64)(44,65)(46,66)(50,67)(51,69)(52,68)(57,70)(58,71)(60,72);(9,13,14,31,15,32,22,46)(10,18,19,41,20,30,33,58)(11,24,26,52,25,51,34,60)(12,27,29,55,28,54,45,66)(16,36,21,43,37,57,44,50)(17,38,40,63,39,53,56,71)(23,47,48,68,49,69,59,72)(35,61,42,64,62,70,65,67);(11,25)(12,28)(16,37)(17,39)(21,44)(24,51)(26,34)(27,54)(29,45)(36,57)(38,53)(40,56)(43,50)(52,60)(55,66)(63,71)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,e^b=ei,e^c=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536911);