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)(7,14,16,20)(8,12)(9,18)(10,13)(11,17,19,15)(21,22)(23,26)(24,31,33,27)(25,28)(29,36,32,34)(30,35);(1,2,3,4)(7,15,16,17)(11,14,19,20)(21,23)(22,26)(24,32,33,29)(25,30)(27,34,31,36)(28,35);(5,7)(6,11)(8,15)(9,17)(10,16)(12,20)(13,19)(14,18)(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,f^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23440);