Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2847 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,24,30,25,29,31,35)(22,26,27,33,28,32,34,36);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,29)(26,32)(30,35)(33,36);(6,14)(11,15)(12,13)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6295352);