Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164512 |
| 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 | #36144 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,19,20,17)(14,18)(21,22)(23,30,32,35)(24,33)(25,28)(26,29)(27,31,36,34);(1,2)(3,4)(5,7,9,14)(6,10,12,18)(8,13,16,20)(11,17,15,19);(5,8,9,16)(6,11,12,15)(7,13,14,20)(10,17,18,19)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(2,4)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,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, 128946);