Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13396 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,16,18,32)(8,22)(9,14)(10,28)(11,15)(12,19,29,34)(13,26)(17,33,31,24)(20,30,21,35)(23,25)(27,36);(1,2)(3,4)(5,6)(7,17)(8,23)(9,28)(10,14)(11,15)(12,20)(13,27)(16,24)(18,31)(19,35)(21,29)(22,25)(26,36)(30,34)(32,33);(1,3,4,2)(5,7,10,20,11,18,9,21)(6,12,14,31,15,29,28,17)(8,24,26,19,27,33,25,34)(13,30,22,16,23,35,36,32)Polycyclic presentation:
2^9|a^2=g,c^2=dh,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320182, #72320289, #72320481, … |
| maximal normal subgroup of | #72320182, #72320289, #72320481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55833);