Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,14,9,23)(7,16,18,12,19,28,24,27)(8,13,21,32,22,30,25,31)(17,33,34,26,35,29,20,36);(5,7)(6,12)(8,20)(9,24)(10,18)(11,19)(13,29)(14,27)(15,28)(16,23)(17,21)(22,34)(25,35)(26,31)(30,33)(32,36);(2,4);(5,8)(6,13)(7,17)(9,25)(10,21)(11,22)(12,26)(14,30)(15,32)(16,33)(18,34)(19,35)(20,24)(23,31)(27,36)(28,29);(6,14)(12,27)(13,30)(15,23)(16,28)(26,36)(29,33)(31,32)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,f^b=fi,f^d=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250524, #72250532, #72250537, … |
| maximal normal subgroup of | #72250524, #72250532, #72250537, … |
| maximal quotient of | #72318429, D8×#407, C2×#6379174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286370);