Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11)(22,27,28,31)(23,24)(25,33)(29,36,35,30)(32,34);(2,4)(5,7)(6,11,13,19)(8,16)(9,17)(10,15)(12,14,18,20)(21,22,24,29,33,36,23,31,26,28,34,35,25,30,32,27);(7,15)(11,19)(14,20)(16,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgi,g^a=gi,c^b=ch,d^b=di,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225030, #72225090, #72225106, … |
| maximal normal subgroup of | #72225030, #72225090, #72225106, … |
| maximal quotient of | #72225028, #72286517, C2×#126846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34042);