Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,14,16,10)(8,13)(9,12)(11,15,20,17)(18,19);(5,7)(6,10)(8,17)(9,16)(11,19)(12,14)(13,20)(15,18);(1,2,3,4)(5,7)(6,11,12,20)(8,15)(9,16)(10,13,14,19)(17,18);(5,7)(6,10)(8,15)(9,16)(11,13)(12,14)(17,18)(19,20)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,h^a=hi,c^b=ce,f^b=fi,g^b=gi,g^c=gi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72225792, #72225797, #72225802, … |
| maximal normal subgroup of | #72225792, #72225797, #72225802, … |
| maximal quotient of | #72225787, #72225788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263131);