Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,11,17,12,18,10,16)(7,19,20,33,22,32,21,13)(8,14,25,35,26,31,24,34)(9,27,23,15,28,36,29,30);(5,7)(6,13)(8,23)(9,25)(10,20)(11,21)(12,22)(14,27)(15,34)(16,19)(17,32)(18,33)(24,28)(26,29)(30,35)(31,36);(2,4)(5,8)(6,14)(7,9)(10,24)(11,25)(12,26)(13,30)(15,33)(16,34)(17,35)(18,31)(19,27)(20,23)(21,29)(22,28)(32,36);(2,4)(5,9)(6,15)(7,8)(10,23)(11,29)(12,28)(13,31)(14,33)(16,36)(17,27)(18,30)(19,35)(20,24)(21,25)(22,26)(32,34);(5,10,12,11)(6,16,18,17)(7,20,22,21)(8,24,26,25)(9,23,28,29)(13,19,33,32)(14,34,31,35)(15,36,30,27)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,f^b=fi,h^b=hi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408103);