Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| socle | C2 | C22×AΣL(1,𝔽16) |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,9,19,27)(3,13,16,23,17,29,31,8)(4,10,18,11,6,20,22,26)(14,30,15,25,21,28,32,24);(1,3)(2,8)(4,14)(5,15)(6,21)(7,17)(9,23)(10,24)(11,13)(12,28)(16,22)(18,31)(19,32)(20,25)(26,29)(27,30);(2,9)(8,23)(10,20)(11,26)(12,27)(13,29)(24,25)(28,30);(1,4,7,6)(2,10,9,20)(3,14,17,21)(5,18,19,22)(8,24,23,25)(11,27,26,12)(13,30,29,28)(15,31,32,16)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gh,h^a=hi,f^b=fgi,g^b=gi,h^b=hi,f^e=fhi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279249, #72279299, #72279638, … |
| maximal normal subgroup of | #72279249, #72279299, #72279638, … |
| maximal quotient of | #72279286, #72340602, #72340662 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419767);