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 | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,16)(7,19)(8,10,17,27)(9,15,12,28)(13,25,26,18)(14,22)(20,32)(21,33)(23,36)(24,30)(29,34)(31,35);(1,2,3,4)(5,7,12,22,13,23,10,21,11,24,9,20,26,34,27,35)(6,14,17,30,18,31,28,36,16,32,8,19,25,33,15,29);(5,8,9,18,13,15,27,16,11,17,12,25,26,28,10,6)(7,19,20,31,23,29,35,32,24,30,22,33,34,36,21,14)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,f^b=fi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250565, #72250572, #72250583, … |
| maximal normal subgroup of | #72250565, #72250572, #72250583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54127);