Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
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×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,21,23,13)(9,24,18,32)(10,26)(11,16)(12,19)(14,25,29,27)(15,28)(20,34)(22,31,35,33)(30,36);(1,2)(3,4)(5,7,10,19,11,20,28,36)(6,12,15,17,16,30,26,34)(8,9,22,25,23,18,35,27)(13,14,31,32,21,29,33,24);(2,4)(5,8)(6,13)(7,18)(9,20)(10,22)(11,23)(12,29)(14,30)(15,31)(16,21)(17,24)(19,27)(25,36)(26,33)(28,35)(32,34);(5,9,11,18)(6,14,16,29)(7,8,20,23)(10,27,28,25)(12,13,30,21)(15,24,26,32)(17,33,34,31)(19,35,36,22)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=hi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290413);