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) = C24×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 | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,11,15,28,31,12,14,29,33,9,26,22,35)(7,17,19,34,18,23,13,30,20,8,21,24,27,25,32,36);(1,2,3,4)(5,7)(6,8)(9,27)(10,21)(11,18)(12,20)(13,22)(14,17)(15,25)(16,34)(19,29)(23,26)(24,33)(28,32)(30,31)(35,36);(5,8,22,30,9,23,29,34,12,17,28,36,11,25,10,24)(6,13,31,27,26,19,16,20,14,32,35,18,15,21,33,7)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,e^b=ei,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239414, #72239420, #72239424, … |
| maximal normal subgroup of | #72239414, #72239420, #72239424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54558);