Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35256 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,24)(19,26)(20,28)(21,22)(25,27)(31,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,24,23,26,22,19)(20,25,29,31,28,32,30,27);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,19,21,18,23,24,22,26)(20,27,29,25,28,31,30,32);(10,14)(13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32);(20,28)(25,32)(27,31)(29,30)Polycyclic presentation:
2^9|b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340948);