Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C2×#1889 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,17,23)(3,11,15,25,13,24,28,32)(4,8,12,19,16,22,26,30)(7,18,21,27,20,29,31,14);(1,3,12,18,6,13,26,29)(2,7,19,25,10,20,30,32)(4,14,5,15,16,27,17,28)(8,11,9,21,22,24,23,31);(4,16)(5,17)(14,27)(15,28);(3,13)(7,20)(11,24)(14,27)(15,28)(18,29)(21,31)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fhi,e^c=eh,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231746, #72236796, #72236884, … |
| maximal normal subgroup of | #72231746, #72236884, #72237529, … |
| maximal quotient of | #72267968, C2×#494587, #72278460, … |
| automorphism group of | #1328 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401783);