Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,10,16,9,24)(7,17,20,12,21,30,25,29)(8,13,22,14,23,32,26,31)(18,33,35,27,19,28,34,36);(1,2,3,4)(5,7,9,25,10,21,11,20)(6,12,24,17,16,29,15,30)(8,19,26,35,23,18,22,34)(13,27,31,33,32,36,14,28);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,26)(12,27)(14,24)(15,31)(16,32)(17,28)(19,21)(20,34)(25,35)(29,36)(30,33);(5,8)(6,14)(7,19)(9,26)(10,23)(11,22)(12,28)(13,24)(15,32)(16,31)(17,27)(18,21)(20,34)(25,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273921);