Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #2676 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1729 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,27,32,26,29,24,30)(23,25,31,36,33,28,34,35);(5,7,10,17)(6,11,14,15)(8,16,9,18)(12,19,13,20)(21,23,26,33)(22,28,29,25)(24,34,27,31)(30,36,32,35);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(22,29)(23,33)(30,32)(31,34);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24,26,27)(22,30,29,32)(23,31,33,34)(25,36,28,35);(6,14)(11,15)(12,13)(19,20)(21,25,27,36,26,28,24,35)(22,31,32,33,29,34,30,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=df,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311673);