Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,16,11,17,10,15)(7,18,20,13,21,29,27,12)(8,22,25,33,26,14,24,32)(19,34,23,28,36,31,35,30);(1,2)(3,4)(5,7)(6,12)(8,23)(9,27)(10,20)(11,21)(13,17)(14,31)(15,18)(16,29)(19,25)(22,34)(24,36)(26,35)(28,32)(30,33);(2,4)(5,7)(6,13)(8,19)(9,20)(10,27)(11,21)(12,17)(14,30)(15,18)(16,29)(22,28)(23,25)(24,35)(26,36)(31,33)(32,34);(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,28)(13,30)(15,32)(16,33)(17,22)(18,34)(20,35)(21,36)(23,27)(29,31);(5,9,11,10)(6,15,17,16)(7,20,21,27)(8,24,26,25)(12,29,13,18)(14,32,22,33)(19,35,36,23)(28,31,30,34)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408088);