Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,17,9,15)(7,12,19,28,20,14,25,33)(8,21,23,36,24,27,26,29)(13,22,31,18,32,34,30,35);(1,2,3,4)(5,7)(6,12)(8,22)(9,25)(10,19)(11,20)(13,29)(14,17)(15,33)(16,28)(18,23)(21,31)(24,34)(26,35)(27,30)(32,36);(5,8,11,24)(6,13,17,32)(7,18,20,35)(9,26,10,23)(12,27,14,21)(15,31,16,30)(19,34,25,22)(28,36,33,29)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,f^c=fi,g^c=gh,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55202);