Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10254 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,7,5,3,8,6)(9,10,12,16,14,11,13,15);(1,3)(2,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16);(10,15)(11,16)(12,13)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,f^b=fi,d^c=di,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214287, #72214564, #72214595, … |
| maximal normal subgroup of | #72214287, #72214564, #72214595, … |
| maximal quotient of | #72215787, C2×#115568 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22764);