Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,11)(7,12)(8,9)(13,14)(15,19)(16,18)(17,20);(5,6,8,11,10,12,9,7)(13,15)(14,17,18,19)(16,20);(2,4)(5,7,8,6,10,11,9,12)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72258163, #72258174, #72258197, … |
| maximal normal subgroup of | #72258163, #72258174, #72258197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17623);