Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,16)(14,19)(15,17)(18,20)(21,22,25,29)(23,30,32,27)(24,28,26,34)(31,36,33,35);(6,10,12,13)(11,17,19,18)(21,23,26,33,25,32,24,31)(22,27,28,35,29,30,34,36);(1,3,4,2)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(6,12)(7,14)(10,13)(15,20)(21,24,25,26)(22,28,29,34)(23,31,32,33)(27,35,30,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=be,d^a=dh,e^a=eh,g^a=ghi,e^b=ei,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90441);