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×C4 |
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×#380 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1482 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)(7,15)(8,19,21,12)(9,22)(10,14)(11,18)(13,23)(16,27)(17,31,29,28)(20,26,35,34)(24,30)(25,32,33,36);(5,7,9,18,10,16,23,30)(6,11,13,15,14,24,22,27)(8,17,20,32,21,29,35,36)(12,25,26,28,19,33,34,31);(1,2,3,4)(7,16)(11,24)(15,27)(17,29)(18,30)(25,33)(28,31)(32,36);(5,8)(6,12)(7,17)(9,20)(10,21)(11,25)(13,26)(14,19)(15,28)(16,29)(18,32)(22,34)(23,35)(24,33)(27,31)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,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, 299661);