Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,18)(17,20);(6,10,11,12)(13,15)(14,17,18,19)(16,20);(1,3,4,2)(6,11)(10,12);(5,7,9,8)(6,12,11,10)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,b^a=be,c^a=ch,e^a=eh,g^a=ghi,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72309174, #72309181, #72309244, … |
| maximal normal subgroup of | #72309174, #72309181, #72309244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89792);