Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,11,12,9)(8,10)(13,14,17,20,16,19,18,15);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,20);(13,15,18,19,16,20,17,14)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,e^2=i,g^2=i,b^a=bd,d^a=dh,c^b=ci,e^b=eh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72313490, #72313509, #72313525, … |
| maximal normal subgroup of | #72313490, #72313509, #72313525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12501);