Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17)(8,19)(9,24)(10,14)(11,15)(12,16)(13,28)(18,30)(20,22)(21,25)(23,29)(26,32)(27,31)(33,36)(34,35);(1,2)(3,4)(5,7)(6,8)(9,25)(10,19)(11,29)(12,20)(13,30)(14,22)(15,17)(16,23)(18,27)(21,31)(24,35)(26,34)(28,33)(32,36);(2,4)(5,8)(6,7)(9,21)(10,22)(11,17)(12,23)(13,18)(14,19)(15,29)(16,20)(24,34)(25,31)(26,35)(27,30)(28,36)(32,33);(5,9,11,27,12,28,10,26)(6,13,14,31,16,24,15,32)(7,18,19,25,20,34,29,33)(8,21,17,30,23,36,22,35);(5,10,12,11)(6,14,16,15)(7,19,20,29)(8,22,23,17)(9,26,28,27)(13,31,24,32)(18,25,34,33)(21,35,36,30)Polycyclic presentation:
2^9|d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409345);