Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,20,25,22,24,21,26)(19,27,29,32,28,23,30,31);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,21,29,22,28,20,30)(18,23,26,31,24,27,25,32);(3,11)(7,15)(10,16)(12,13)(19,28)(23,27)(29,30)(31,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239268, #72239272, #72239273, … |
| maximal normal subgroup of | #72239268, #72239272, #72239273, … |
| maximal quotient of | #72239282, C2×#181290, #72309166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88486);