Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#2200 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,14)(7,12,8,13)(9,11,17,20)(15,18,16,19);(6,11)(7,8)(12,19)(13,18)(14,20)(15,16);(5,7,10,8)(6,12,14,13)(9,16,17,15)(11,18,20,19);(1,3,4,2)(5,8,10,7)(6,13,14,12)(9,16,17,15)(11,19,20,18)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=g,b^a=be,c^a=cf,h^a=hi,c^b=ci,d^b=di,f^b=fhi,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72278265, #72281265, C4×#33553, … |
| maximal normal subgroup of | #72278265, #72281265, C4×#33553, … |
| maximal quotient of | #72279017, #72283097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402972);