Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17)(8,21)(9,16)(10,15)(11,14)(12,19)(13,24)(18,29)(20,31)(22,27)(23,30)(25,28)(26,32)(33,36)(34,35);(1,2)(3,4)(5,7)(6,12)(8,22)(9,25)(10,19)(11,20)(13,29)(14,28)(15,31)(16,17)(18,23)(21,35)(24,33)(26,34)(27,30)(32,36);(2,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,27)(14,28)(15,17)(16,31)(21,36)(22,23)(24,34)(26,33)(29,30)(32,35);(5,8,9,23,11,24,10,26)(6,13,15,30,14,21,16,32)(7,18,19,22,20,34,25,33)(12,27,17,29,28,36,31,35);(6,14)(12,28)(13,21)(15,16)(17,31)(27,36)(29,35)(30,32)Polycyclic presentation:
2^9|d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409346);