Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(6,11,14,23)(7,15,19,18)(8,20,22,21)(26,32,35,36);(1,2,4,3)(5,6,9,23,10,14,24,11)(7,16,18,12,19,27,15,28)(8,13,21,29,22,30,20,25)(17,32,33,26,34,36,31,35);(5,7,9,18,10,19,24,15)(6,12,11,16,14,28,23,27)(8,17,21,33,22,34,20,31)(13,26,25,32,30,35,29,36);(1,3,4,2)(5,8,10,22)(6,13,14,30)(7,17,19,34)(9,21,24,20)(11,25,23,29)(12,26,28,35)(15,31,18,33)(16,32,27,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277027);