Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2428 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1805 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,28,23,27,32,26,31)(24,29,25,30,35,33,36,34);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23,28,32,27,31,26,22)(24,34,25,29,35,30,36,33);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,27,35)(22,29,32,33)(23,30,31,34)(25,26,36,28);(7,16)(11,14)(15,17)(19,20)(21,25)(22,30)(23,33)(24,26)(27,36)(28,35)(29,31)(32,34);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(21,26,27,28)(22,31,32,23)(24,25,35,36)(29,30,33,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=df,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296911);