Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,24,30)(7,12,18,9,14,23,29,22)(8,13,21,28,20,27,34,36)(17,31,32,33,26,25,35,19);(1,2)(3,4)(5,7,10,18,11,14,24,29)(6,12,15,9,16,23,30,22)(8,17,21,32,20,26,34,35)(13,25,28,19,27,31,36,33);(2,4)(5,8)(6,13)(7,17)(9,19)(10,21)(11,20)(12,25)(14,26)(15,28)(16,27)(18,32)(22,33)(23,31)(24,34)(29,35)(30,36);(5,9)(6,14)(7,16)(8,19)(10,23)(11,22)(12,24)(13,26)(15,29)(17,27)(18,30)(20,33)(21,31)(25,34)(28,35)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250427, #72252700, #72252977, … |
| maximal normal subgroup of | #72250427, #72252700, #72252977, … |
| maximal quotient of | #72314526, C2×#364517, #72343102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271713);