Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,14,9,23)(7,12,19,26,17,27,18,33)(8,20,21,13,22,29,24,28)(16,30,32,25,31,35,34,36);(2,3)(5,7,10,19,11,17,9,18)(6,12,15,26,14,27,23,33)(8,16,21,32,22,31,24,34)(13,25,29,35,28,36,20,30);(5,8)(6,13)(7,16)(9,24)(10,21)(11,22)(12,25)(14,28)(15,29)(17,31)(18,34)(19,32)(20,23)(26,35)(27,36)(30,33);(1,3,4,2);(6,14)(7,17)(13,28)(15,23)(16,31)(18,19)(20,29)(32,34)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,e^b=ei,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2957 |
| maximal normal subgroup of | D4×#2957 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289099);