Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814811 |
| 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 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 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)(3,7)(4,6)(5,8)(9,10,15,21)(11,16,26,39)(12,27,29,44)(13,18,35,48)(14,20,33,19)(17,40,42,31)(22,41,50,65)(23,37,54,63)(24,49,25,38)(28,55,52,36)(30,56,58,69)(32,45,53,70)(34,47,61,46)(43,64,67,60)(51,66,59,72)(57,68,71,62);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,28)(13,32)(14,24)(15,26)(17,41)(18,45)(19,38)(20,49)(21,39)(22,31)(23,34)(25,33)(27,55)(29,52)(30,59)(35,53)(36,44)(37,47)(40,50)(42,65)(43,68)(46,63)(48,70)(51,58)(54,61)(56,72)(57,60)(62,67)(64,71)(66,69);(2,6)(3,8)(9,12,15,29)(10,17)(11,22)(13,30,35,58)(14,31,33,40)(16,36,39,55)(18,43)(19,27)(20,44)(21,42)(23,51)(24,52)(25,28)(26,50)(32,57)(34,60,61,64)(37,62,63,68)(38,41,49,65)(45,66,70,72)(46,56)(47,69)(48,67)(53,71)(54,59);(2,6)(5,7)(12,29)(17,42)(22,50)(27,44)(28,52)(30,58)(31,40)(36,55)(41,65)(43,67)(51,59)(56,69)(57,71)(60,64)(62,68)(66,72);(9,13,33,61,15,35,14,34)(10,18,19,46,21,48,20,47)(11,23,24,53,26,54,25,32)(12,30,40,64,29,58,31,60)(16,37,49,70,39,63,38,45)(17,43,27,56,42,67,44,69)(22,51,52,71,50,59,28,57)(36,62,65,72,55,68,41,66)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,e^2=fhi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723633);