Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(9,10,15,16)(11,13,12,14);(5,6)(7,8)(9,11,14,10,15,12,13,16);(2,4)(6,8)(9,12,14,16,15,11,13,10);(6,8);(9,13,15,14)(10,12,16,11)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239649, #72281437, #72281542, … |
| maximal normal subgroup of | #72239649, #72281437, #72281542, … |
| maximal quotient of | #72239687, #72239694, #72305640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346168);