Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433040 |
| 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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2222 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)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(1,2,3,4)(7,13)(10,16)(14,19)(17,20);(6,11)(10,16)(12,18)(17,20)(23,30)(26,35)(29,36)(31,34);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,30)(26,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344421);