Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,13,15,20)(8,10)(9,12)(11,14,18,17)(16,19)(21,22,24,29,33,36,32,27,26,31,23,28,25,30,34,35);(1,2,3,4)(6,10)(9,16)(11,18)(14,17)(22,27)(23,32)(24,34)(25,33)(28,30)(29,36)(31,35);(5,7,8,15)(6,11,10,18)(9,17,16,14)(12,20,19,13)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,c^2=h,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44193);