Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#354 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| socle | C2 | C22×#354 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
837 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)(3,13,17,24)(4,9)(5,20)(6,11,22,26)(7,12)(8,16,25,32)(10,21,28,19)(14,27,18,30)(15,29,31,23);(1,3,7,14)(2,8)(4,15,5,16)(6,17,22,18)(9,24)(10,27)(11,29)(12,23)(13,28)(19,31,21,32)(20,30)(25,26);(3,14)(8,23)(13,27)(15,16)(17,18)(24,30)(25,29)(31,32);(1,4,7,5)(2,9,12,20)(3,15,14,16)(6,19,22,21)(8,24,23,30)(10,26,28,11)(13,29,27,25)(17,31,18,32)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^b=ehi,f^b=fghi,g^b=gi,f^e=fhi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279298, #72279299, #72279311, … |
| maximal normal subgroup of | #72279298, #72279299, #72279311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419831);