Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #2407 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,15)(9,17)(10,13)(12,14)(16,19)(18,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(1,2)(3,4)(5,7,10,9)(6,11,13,17)(8,14,16,18)(12,19,20,15)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8)(6,12)(7,14)(9,18)(10,16)(11,19)(13,20)(15,17)(23,29)(26,32)(30,35)(33,36);(2,4)(5,7,10,9)(6,11,13,17)(8,14,16,18)(12,19,20,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,f^c=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128689);