Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,10)(7,8)(11,12)(13,14,17,20)(15,19,16,18);(5,6,8,12,9,11,7,10)(14,18)(15,16)(19,20);(1,3,4,2)(6,11)(10,12)(13,15,17,16)(14,19,20,18);(5,7,9,8)(6,10,11,12)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=h,g^2=hi,b^a=be,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279172, #72279570, #72279671 |
| maximal normal subgroup of | #72279172, #72279570, #72279671 |
| maximal quotient of | #72279558, #72309626, #72309653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89989);