Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201948 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,13,3,8)(4,16,19,29,18,27,15,22)(7,10,20,26,21,28,14,23)(9,25,24,32,12,31,30,17);(2,8,13,11)(4,17)(9,22,12,29)(10,23,28,26)(15,31)(16,30,27,24)(18,32)(19,25);(2,9)(3,14)(4,18)(5,20)(8,22)(10,16)(11,29)(12,13)(15,25)(17,32)(19,31)(23,30)(24,26)(27,28);(2,10)(8,23)(9,16)(11,26)(12,27)(13,28)(22,30)(24,29)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,f^b=fi,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72210564, #72213569, #72213681, … |
| maximal normal subgroup of | #72210564, #72213569, #72213681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109220);