Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7503371 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2804 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,12,7,13,6,11)(3,14,22,26,17,25,16,8)(4,10,20,27,19,28,21,9)(15,30,32,24,31,23,18,29);(1,3)(2,8)(4,18)(5,22)(6,16)(7,17)(9,24)(10,23)(11,25)(12,14)(13,26)(15,20)(19,32)(21,31)(27,29)(28,30);(1,4)(2,9,13,27)(3,15,17,31)(5,20)(6,21)(7,19)(8,23)(10,11,28,12)(14,29)(16,18,22,32)(24,25)(26,30);(1,4)(2,10)(3,15,17,31)(5,20)(6,21)(7,19)(8,24,26,29)(9,11)(12,27)(13,28)(14,23,25,30)(16,18,22,32);(4,19)(9,27)(10,28)(15,31)(18,32)(20,21)(23,30)(24,29)Polycyclic presentation:
2^9|a^2=fgh,c^2=gh,d^2=fg,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338916, #72338920, #72338924, … |
| maximal normal subgroup of | #72338916, #72338920, #72338924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421154);