Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11915 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1533 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| socle | C2 | C2×#1533 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,4,9,17,26,6,8,20,25,18,10,27,28)(3,12,14,30,15,7,21,23,16,24,31,29,13,22,19,32);(1,3)(2,7)(4,13)(5,19)(6,16)(8,22)(9,12)(10,24)(11,29)(14,27)(15,18)(17,31)(20,21)(23,26)(25,30)(28,32);(2,8)(5,20)(7,22)(9,10)(12,24)(14,31)(17,27)(19,21);(1,4,6,18)(2,9,8,10)(3,13,16,15)(5,17,20,27)(7,12,22,24)(11,26,25,28)(14,19,31,21)(23,30,32,29)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=h,f^2=g,g^2=i,b^a=be,c^a=ch,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,f^c=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72306758, #72306770, #72306968, … |
| maximal normal subgroup of | #72306758, #72306770, #72306968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390216);