Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,11)(7,15)(9,21)(12,26)(13,29)(14,25)(16,32)(17,22)(18,33)(19,35)(23,27)(24,31)(28,36)(30,34);(5,6,8,11)(7,12,16,23)(9,13,19,24)(10,14,20,25)(15,27,32,26)(17,28,18,34)(21,31,35,29)(22,30,33,36);(5,7,10,17)(6,12,14,28)(8,18,20,16)(9,22,21,15)(11,23,25,34)(13,30,31,27)(19,32,35,33)(24,36,29,26)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,e^c=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55449);