Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,18)(8,21,24,19)(9,15)(10,28)(11,20)(12,17)(13,23,30,34)(14,27)(16,26)(22,31,32,33)(25,29,36,35);(5,7,10,20,11,14,9,6,12,15,27,16,26,28,18,17)(8,19,23,33,25,29,22,13,24,21,34,31,36,35,32,30);(1,2,3,4)(5,8,10,23,11,25,9,22,12,24,27,34,26,36,18,32)(6,13,15,21,16,31,28,35,17,30,7,19,20,33,14,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239801, #72239813, #72239817, … |
| maximal normal subgroup of | #72239801, #72239813, #72239817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54282);