Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32659 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2961 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,10,5,4,6,12)(3,13,16,9,17,25,20,15)(7,11,23,26,19,18,21,27)(14,28,30,24,31,29,22,32);(3,14,17,31)(6,8)(7,21)(9,24,15,32)(10,12)(11,27)(13,29,25,28)(16,22,20,30)(18,26)(19,23);(1,3)(2,9)(4,15)(5,17)(6,20)(7,22)(8,16)(10,25)(11,28)(12,13)(14,23)(18,29)(19,30)(21,31)(24,26)(27,32);(1,4,8,12,5,2,6,10)(3,15,16,13,17,9,20,25)(7,11,23,26,19,18,21,27)(14,24,30,29,31,32,22,28)Polycyclic presentation:
2^9|a^2=ef,b^2=efh,d^2=ef,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi,g^d=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#511541, #72320511, … |
| maximal normal subgroup of | #72237280, C2×#511541, #72320511, … |
| maximal quotient of | C2×#511541, #72250132, #72250133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418737);