Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(9,10)(11,20)(12,16)(13,28)(14,19)(15,22)(17,25)(18,26)(21,36)(23,38)(24,35)(27,33)(29,34)(30,37)(31,39)(32,40);(2,4)(5,6)(7,8)(9,11)(10,15)(12,24)(13,29)(14,21)(16,20)(17,34)(18,35)(19,30)(22,25)(23,26)(27,39)(28,31)(32,38)(33,36)(37,40);(6,8)(10,16)(12,25)(13,26)(15,20)(17,18)(19,33)(22,24)(23,29)(27,40)(28,32)(30,36)(31,38)(34,35)(37,39);(11,21)(15,30)(20,36)(22,37)(23,29)(24,39)(31,35)(34,38)Polycyclic presentation:
2^9|e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319972, #72320769, #72320810, … |
| maximal normal subgroup of | #72319972, #72320769, #72320810, … |
| maximal quotient of | #72322619, #72322958, #72322968 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406041);