Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19254 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,7,5,8,3,6)(9,10,12,14,13,15,11,16);(2,6,8,7)(10,14)(11,12)(15,16);(10,14)(11,12)(15,16);(10,15)(14,16)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214287, #72214454, #72214520, … |
| maximal normal subgroup of | #72214287, #72214454, #72214520, … |
| maximal quotient of | #72215791, C2×#198038, #72257820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105234);