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 | |
3095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,15,11,16,10,14)(7,17,19,12,21,32,20,27)(8,13,25,29,24,31,23,30)(18,28,34,26,35,33,22,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,20)(10,19)(11,21)(13,28)(14,32)(15,17)(16,27)(18,23)(24,34)(25,35)(26,30)(29,36)(31,33);(2,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,32)(16,27)(22,23)(24,35)(25,34)(28,30)(29,33)(31,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,30)(16,31)(17,33)(19,22)(20,34)(21,35)(27,36)(28,32);(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,24,25)(12,17,27,32)(13,29,31,30)(18,22,35,34)(26,33,36,28)Polycyclic presentation:
2^9|a^2=fi,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, 7408089);