Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
848 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)(7,13,15,19)(8,11)(9,12)(10,16,14,17)(18,20)(21,22)(23,26)(24,31,33,27)(25,28)(29,34,32,36)(30,35);(1,2,3,4)(5,6)(7,14,15,10)(8,11)(9,12)(13,16,19,17)(18,20)(21,23)(22,26)(24,32,33,29)(25,30)(27,36,31,34)(28,35);(5,7)(6,10)(8,17)(9,15)(11,19)(12,14)(13,20)(16,18)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=i,e^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ehi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23311);