Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #165149 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36147 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13,14,10)(8,11)(9,12)(15,19,20,17)(16,18)(21,22)(23,26)(24,27)(25,28)(29,32)(30,34)(31,35)(33,36);(1,2)(3,4)(7,14)(10,13)(15,20)(17,19)(21,23)(22,26)(24,31)(25,30)(27,35)(28,34)(29,33)(32,36);(5,7)(6,10)(8,15)(9,14)(11,17)(12,13)(16,20)(18,19)(22,27)(23,30)(24,32)(26,33)(28,29)(34,35);(2,4)(5,8,9,16)(6,11,12,18)(7,15,14,20)(10,17,13,19)(22,28)(26,34)(27,29)(33,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128967);