Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C4×B(4,𝔽2) |
| prime radical | #33546 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,16,22)(3,8,12,20,13,21,25,30)(4,7,15,19,14,18,26,29)(11,17,24,28,23,27,31,32);(2,7)(8,17)(9,19)(10,18)(20,28)(21,27)(22,29)(30,32);(4,14)(11,23)(15,26)(24,31);(1,3,5,12,6,13,16,25)(2,8,9,20,10,21,22,30)(4,11,15,24,14,23,26,31)(7,17,19,28,18,27,29,32)Polycyclic presentation:
2^9|a^2=g,d^2=g,g^2=i,b^a=be,c^a=cf,h^a=hi,f^b=fhi,e^c=eh,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231703, #72236846, #72236884, … |
| maximal normal subgroup of | #72231703, #72236884, #72237525, … |
| maximal quotient of | #72257245, C2×#495753, #72312879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402949);