Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,30)(23,25,29,26)(24,32,35,31)(28,33,36,34);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,30,27,29,26,22)(24,32,33,28,35,31,34,36);(7,13)(10,17)(14,19)(18,20)(21,24,25,33,27,35,26,34)(22,28,23,31,30,36,29,32);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,29,30,23)(24,33,35,34)(28,32,36,31);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,30)(23,29)(28,36)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404890);