Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(6,8)(10,12)(13,14)(15,18)(16,19)(17,20);(2,3)(10,12)(13,15,16,20)(14,17,19,18);(9,10,11,12)(13,14)(15,18)(16,19)(17,20);(2,3)(5,6)(7,8)(13,14)(15,17)(16,19)(18,20);(1,3,4,2)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72232413, #72232439, #72232470, … |
| maximal normal subgroup of | #72232413, #72232439, #72232470, … |
| maximal quotient of | #72298260, #72298361, #72330405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 692805);