Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,23)(10,13)(11,15)(12,17)(14,25)(18,29)(19,33)(20,26)(22,28)(24,35)(27,34)(30,32)(31,36);(5,7,9,19,10,20,24,34,11,18,25,32,8,17,22,31)(6,12,14,27,13,16,28,30,15,26,23,36,21,29,35,33);(1,3,4,2)(5,6)(7,12)(8,21)(9,14)(10,13)(11,15)(16,20)(17,29)(18,26)(19,27)(22,35)(23,25)(24,28)(30,34)(31,33)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239196, #72239201, #72239420, … |
| maximal normal subgroup of | #72239196, #72239201, #72239420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54314);