Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6852 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)(3,4)(5,6)(7,8)(9,10,12,11)(13,14)(15,18)(16,17)(19,20);(2,4)(5,6)(7,8)(9,11,12,10);(6,8)(14,17)(18,20);(2,4);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch152 conjugacy classes — view the full character table.
| maximal subgroup of | #72239451, #72239620, #72249699, … |
| maximal normal subgroup of | #72239451, #72239620, #72249699, … |
| maximal quotient of | #72239413, #72249752, #72251003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421718);