Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483425 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,11,16,10,25)(7,13,19,29,17,27,20,35)(8,21,23,30,24,12,22,14)(18,32,34,36,31,26,33,28);(6,12,16,21)(8,22)(9,10)(13,26,27,32)(14,15,30,25)(18,33)(19,20)(23,24)(28,29,36,35)(31,34);(2,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,26)(14,28)(15,29)(16,27)(21,32)(22,33)(23,34)(24,31)(25,35)(30,36);(7,17)(13,27)(18,31)(19,20)(26,32)(28,36)(29,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231484, #72231551, #72231555, … |
| maximal normal subgroup of | #72231484, #72231551, #72231555, … |
| maximal quotient of | #72257056, C2×#494267, #72309689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401463);