Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 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)(21,22,27,35,26,29,25,24)(23,30,36,28,31,34,33,32);(7,13)(8,16)(10,17)(11,15)(21,23)(22,28)(24,34)(25,36)(26,31)(27,33)(29,32)(30,35);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(22,29)(23,31)(24,35)(33,36);(7,13)(10,17)(14,19)(18,20)(21,24,25,29,26,35,27,22)(23,32,33,34,31,28,36,30);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(23,31)(24,35)(33,36)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104056);