Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| 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 | |
956 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)(5,6,10,15,12,8,11,16)(7,17,21,35,22,36,20,32)(9,24,26,14,28,33,27,23)(13,29,31,19,18,25,30,34);(5,7,11,21,12,22,10,20)(6,13,15,30,8,18,16,31)(9,25,27,34,28,29,26,19)(14,32,33,36,23,35,24,17);(2,4)(5,8)(6,12)(7,18)(9,23)(10,15)(11,16)(13,22)(14,28)(17,34)(19,36)(20,30)(21,31)(24,27)(25,35)(26,33)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320894, #72320995, #72321011, … |
| maximal normal subgroup of | #72320894, #72320995, #72321011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53750);