Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18,17,19,15,20);(1,2,3,4)(14,18)(15,16)(19,20);(6,9)(10,12);(5,7,8,11)(6,10,9,12);(14,19)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239081, #72239084, #72239092, … |
| maximal normal subgroup of | #72239081, #72239084, #72239092, … |
| maximal quotient of | #72239122, #72306130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279592);