Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,26,22,35,9,16,10,15,28,32,8,14,25,36)(7,17,24,33,18,31,19,34,21,12,27,13,20,29,23,30);(1,2)(3,4)(5,7)(6,12)(8,20)(9,23)(10,21)(11,27)(13,32)(14,31)(15,17)(16,34)(18,22)(19,25)(24,28)(26,33)(29,35)(30,36);(2,4)(5,7)(6,13,15,33)(8,18)(9,24)(10,21)(11,19)(12,26,17,32)(14,30,35,34)(16,29,36,31)(20,22)(23,28)(25,27);(5,8,10,22)(6,14,15,35)(7,18,21,20)(9,11,25,28)(12,29,17,31)(13,30,33,34)(16,26,36,32)(19,27,23,24)Polycyclic presentation:
2^9|a^2=fh,c^2=e,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320182, #72320223, #72320467, … |
| maximal normal subgroup of | #72320182, #72320223, #72320467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406584);