Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106326 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,20)(3,11)(4,15)(8,18,19,30)(10,24,23,31)(13,27)(14,26)(16,17)(21,22,32,28)(25,29);(1,2,4,10,5,8,16,22)(3,7,13,21,11,18,25,31)(6,9,17,23,12,19,15,28)(14,20,27,32,26,30,29,24);(3,11)(7,18)(13,25)(14,26)(20,30)(21,31)(24,32)(27,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,e^b=eh,f^b=fi,d^c=dh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72210887, #72213377, #72213768, … |
| maximal normal subgroup of | #72210887, #72213377, #72213768, … |
| maximal quotient of | C2×#106328, #72239222, #72240268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13524);