Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
990 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)(3,4)(5,6,9,14,10,15,24,16)(7,17,19,30,21,32,20,12)(8,22,25,13,26,31,34,27)(11,23,28,36,29,35,33,18);(5,7)(6,11,15,29)(8,23)(9,19)(10,21)(12,27,30,13)(14,28,16,33)(17,22,32,31)(18,34)(20,24)(25,36)(26,35);(2,4)(5,7)(6,12)(8,18)(9,20)(10,21)(11,27)(13,29)(14,32)(15,30)(16,17)(19,24)(22,33)(23,34)(25,35)(26,36)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231488, #72231513, #72231521, … |
| maximal normal subgroup of | #72231488, #72231513, #72231521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53753);