Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,21,29,22,28,20,27)(18,23,24,31,25,26,30,32);(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh104 conjugacy classes — view the full character table.
| derived subgroup of | #72235726, C2×#72210004 |
| maximal subgroup of | #72210001, #72210004, #72235049, … |
| maximal normal subgroup of | #72210001, #72210004, #72235049, … |
| maximal quotient of | C2×#181415, #72240045 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88611);