Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814811 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10,14,19)(11,22,25,17)(12,27,29,56)(13,20,34,46)(15,37,16,21)(18,32,45,31)(23,43,51,66)(24,50,53,40)(26,41,38,42)(28,49,57,39)(30,55,59,72)(33,62,54,47)(35,64,36,48)(44,61,68,60)(52,67,71,69)(58,70,63,65);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,33)(14,25)(15,26)(16,38)(18,43)(19,22)(20,47)(21,41)(23,32)(24,35)(27,39)(29,57)(30,52)(31,51)(34,54)(36,53)(37,42)(40,64)(44,65)(45,66)(46,62)(48,50)(49,56)(55,69)(58,60)(59,71)(61,63)(67,72)(68,70);(2,5)(6,8)(9,12,14,29)(10,18)(11,23)(13,30,34,59)(15,31,16,32)(17,39,22,49)(19,45)(20,44)(21,27)(24,52)(25,51)(26,28)(33,63)(35,60,36,61)(37,56)(38,57)(40,65,50,70)(41,66,42,43)(46,68)(47,69,62,67)(48,55)(53,71)(54,58)(64,72);(1,3,4,7)(2,6,5,8)(10,19)(12,29)(17,22)(20,46)(21,37)(23,51)(28,57)(30,59)(31,32)(40,50)(41,42)(47,62)(48,64)(52,71)(58,63)(60,61);(9,13,16,36,14,34,15,35)(10,20,21,48,19,46,37,64)(11,24,26,54,25,53,38,33)(12,30,32,61,29,59,31,60)(17,40,42,62,22,50,41,47)(18,44,27,55,45,68,56,72)(23,52,28,58,51,71,57,63)(39,65,43,67,49,70,66,69)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=fi,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, 6724430);