Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9385 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1126 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)(5,6)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,33)(32,34)(35,36);(5,7)(6,11,13,20)(8,17)(9,15)(10,16)(12,14,18,19)(21,23)(22,26,28,29)(24,30)(25,31)(27,33,34,35)(32,36);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di,e^b=ei,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14273);