Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,15)(8,13)(9,22,20,12)(10,14)(11,17)(16,26)(18,25,34,32)(19,29)(21,28)(23,30,36,27)(24,35,31,33);(1,2,4,3)(5,7,8,17)(6,11,13,15)(9,18,20,34)(10,16,21,19)(12,25,22,32)(14,29,28,26)(23,33,36,35)(24,30,31,27);(1,3,4,2)(6,12)(7,16)(10,21)(11,24)(13,22)(14,30)(15,31)(17,19)(18,33)(23,36)(25,29)(26,32)(27,28)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=di,f^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, 55892);