Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482397 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #35692 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1122 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)(3,14,17,8)(4,19,21,29)(6,13,23,30)(7,10,24,25)(9,22,11,18)(15,27,32,28)(16,31,20,26);(2,8)(4,20)(5,23)(9,25)(10,26)(11,13)(12,29)(14,28)(17,22)(19,27)(24,32)(30,31);(1,3,15,21)(2,9,27,31)(4,5,17,32)(6,18,7,16)(8,25,19,30)(10,29,13,14)(11,28,26,12)(20,23,22,24);(2,10)(3,16)(5,24)(8,26)(11,19)(13,27)(18,21)(23,32)Polycyclic presentation:
2^9|a^2=f,c^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72234703, #72234738 |
| maximal normal subgroup of | #72234703, #72234738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389715);