Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#383 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
802 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)(7,16)(8,20,22,27)(9,14)(10,15)(11,19)(12,21,28,34)(13,23)(17,26)(18,25,30,35)(24,31)(29,32,33,36);(5,7,10,17)(6,11)(8,18,22,30)(9,19,23,31)(12,25)(13,16)(14,26)(15,24)(20,33)(21,32,34,36)(27,29)(28,35);(1,2,3,4)(7,17)(11,24)(16,26)(18,30)(19,31)(25,35)(29,33)(32,36);(5,8,9,21,10,22,23,34)(6,12,13,27,15,28,14,20)(7,18,19,32,17,30,31,36)(11,25,16,29,24,35,26,33)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299722);