Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1999 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3060 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1040 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)(5,6,9,15,11,16,10,28)(7,17,19,33,20,34,27,13)(8,21,24,29,26,35,25,14)(12,22,30,23,31,36,32,18);(5,7)(6,12)(8,22)(9,19)(10,27)(11,20)(13,14)(15,30)(16,31)(17,21)(18,25)(23,24)(26,36)(28,32)(29,33)(34,35);(2,4)(5,7)(6,13)(8,23)(9,27)(10,19)(11,20)(12,29)(14,31)(15,34)(16,33)(17,28)(18,26)(21,30)(22,24)(25,36)(32,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,h^b=hi,d^c=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231488, #72231546, #72231549, … |
| maximal normal subgroup of | #72231488, #72231546, #72231549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54851);