Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,19,15,18,13,27)(11,20,22,16,23,32,28,31)(12,17,25,36,26,34,29,35)(21,37,38,30,39,33,24,40);(2,4)(9,11)(10,16)(12,24)(13,28)(14,22)(15,23)(17,33)(18,31)(19,32)(20,27)(21,25)(26,38)(29,39)(30,35)(34,37)(36,40);(6,8)(9,12)(10,17)(11,21)(13,29)(14,25)(15,26)(16,30)(18,34)(19,36)(20,37)(22,38)(23,39)(24,28)(27,35)(31,40)(32,33);(10,18)(16,31)(17,34)(19,27)(20,32)(30,40)(33,37)(35,36)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320944, #72320961, #72320976, … |
| maximal normal subgroup of | #72320944, #72320961, #72320976, … |
| maximal quotient of | #72321565 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264716);