Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
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 | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,19)(3,12)(4,15)(8,18,11,21)(10,25,24,32)(13,27)(14,30)(16,23)(20,26,31,22)(28,29);(1,2,4,10,5,11,16,26)(3,7,14,20,12,21,28,32)(6,8,15,22,17,9,23,24)(13,18,29,25,27,19,30,31);(3,13)(7,18)(12,27)(14,29)(19,21)(20,25)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,e^b=ei,f^b=fhi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213768, #72213902, #72215564, … |
| maximal normal subgroup of | #72213768, #72213902, #72215564, … |
| maximal quotient of | C2×#108241, #72240403, #72257863 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15437);