Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #476 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1864 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,17,12,18,11,30)(7,19,22,31,24,36,23,35)(8,25,9,29,28,16,27,15)(13,21,32,20,33,26,14,34);(1,2,3,4)(5,7)(6,13)(8,26)(9,20)(10,23)(11,22)(12,24)(14,17)(15,35)(16,19)(18,33)(21,28)(25,36)(27,34)(29,31)(30,32);(5,7)(6,14,18,32)(8,26)(9,20)(10,23)(11,22)(12,24)(13,17,33,30)(15,36,29,19)(16,35,25,31)(21,28)(27,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250461, #72250659, #72250663, … |
| maximal normal subgroup of | #72250461, #72250659, #72250663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54507);