Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| 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 | |
3150 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)(5,6,9,16,10,15,11,12)(7,17,18,26,20,13,25,29)(8,21,22,14,23,27,24,30)(19,33,34,31,35,36,32,28);(6,12)(7,18)(8,22)(9,11)(14,30)(15,16)(20,25)(23,24)(26,29)(28,33)(31,36)(32,34);(2,4)(5,7,10,20)(6,13,15,17)(8,19,23,35)(9,18,11,25)(12,26,16,29)(14,28,30,31)(21,36,27,33)(22,34,24,32);(2,4)(5,8)(6,14)(7,19)(9,24)(10,23)(11,22)(12,27)(13,28)(15,30)(16,21)(17,31)(18,32)(20,35)(25,34)(26,33)(29,36);(6,15)(12,16)(13,17)(14,30)(21,27)(26,29)(28,31)(33,36)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7408106);