Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| 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 | #33519 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1070 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)(5,6,11,17,12,18,8,14)(7,19,20,33,23,34,27,32)(9,15,10,26,25,16,24,31)(13,21,28,22,30,35,29,36);(5,7)(6,13)(8,20)(9,22)(10,21)(11,27)(12,23)(14,28)(15,32)(16,33)(17,29)(18,30)(19,31)(24,35)(25,36)(26,34);(2,4)(5,8,12,11)(6,14,18,17)(7,20,23,27)(9,24,25,10)(13,28,30,29)(15,31,16,26)(19,33,34,32)(21,22,35,36);(5,9,11,10,12,25,8,24)(6,15,17,26,18,16,14,31)(7,21,20,22,23,35,27,36)(13,19,28,33,30,34,29,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,e^2=h,g^2=i,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,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231551, #72249754, #72250863, … |
| maximal normal subgroup of | #72231551, #72249754, #72250863, … |
| maximal quotient of | #72249925, #72256851, #72287141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402923);