Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| 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 | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,16,10,28)(7,17,20,13,21,33,19,34)(8,22,25,29,27,36,26,14)(12,23,31,24,32,35,30,18);(1,2,3,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,29)(14,34)(15,31)(16,32)(17,22)(18,26)(24,25)(27,35)(28,30)(33,36);(5,7)(6,13,16,34)(8,24)(9,19)(10,20)(11,21)(12,29,32,14)(15,17,28,33)(18,27)(22,30,36,31)(23,25)(26,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dh,e^b=ei,h^b=hi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250652, #72250677, #72250684, … |
| maximal normal subgroup of | #72250652, #72250677, #72250684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54254);