Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(9,10,12,14,13,15,11,16);(6,7)(9,10)(11,14)(12,16)(13,15);(1,2,3,4)(5,6,8,7);(5,7,8,6);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224712, #72224719, #72281173, … |
| maximal normal subgroup of | #72224712, #72224719, #72281173, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372743);