Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,18,33)(8,13)(9,22)(10,20)(11,15)(12,17,27,35)(14,24)(19,30,26,34)(21,32)(23,29,36,28)(25,31);(5,7)(6,12)(8,19)(9,23)(10,26)(11,18)(13,29)(14,30)(15,27)(16,31)(17,21)(20,28)(22,33)(24,35)(25,36)(32,34);(2,4)(5,6)(7,12)(8,20)(9,14)(10,13)(11,15)(16,17)(18,27)(19,28)(21,31)(22,24)(23,30)(25,32)(26,29)(33,35)(34,36);(5,8,11,10)(6,13,15,20)(9,24,25,21)(12,27)(14,31,32,22)(17,35)(23,36)(28,29)Polycyclic presentation:
2^9|a^2=g,d^2=gi,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285780, #72320174, #72320189, … |
| maximal normal subgroup of | #72285780, #72320174, #72320189, … |
| maximal quotient of | #72285843, #72285862, #72286924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402920);