Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,26)(10,15)(11,17)(12,18)(13,24)(14,23)(19,30)(20,31)(22,29)(25,36)(27,34)(28,35)(32,33);(5,7)(6,11,15,28)(8,22)(9,18)(10,20)(12,13,30,32)(14,27,26,29)(16,21,31,36)(17,33)(19,23)(24,35)(25,34);(1,3,4,2)(5,7)(6,12)(8,17)(9,19)(10,20)(11,13)(14,31)(15,30)(16,26)(18,23)(21,29)(22,33)(24,34)(25,35)(27,36)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250715, #72250725, #72250729, … |
| maximal normal subgroup of | #72250715, #72250725, #72250729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54423);