Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×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 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,16,23)(3,8,13,21,11,22,25,32)(4,7,15,19,14,18,27,29)(12,17,26,28,24,20,30,31);(2,7)(3,11)(8,20)(9,19)(10,18)(12,24)(13,25)(17,22)(21,31)(23,29)(26,30)(28,32);(4,14)(12,24)(15,27)(26,30);(1,3,5,13,6,11,16,25)(2,8,9,21,10,22,23,32)(4,12,15,26,14,24,27,30)(7,17,19,28,18,20,29,31)Polycyclic presentation:
2^9|a^2=g,d^2=g,g^2=i,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231703, #72236797, #72236849, … |
| maximal normal subgroup of | #72231703, #72236884, #72237521, … |
| maximal quotient of | #72257225, #72312390, #72312494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402975);