Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4276 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)(5,6,8,10)(7,11,12,9)(14,18)(15,16)(19,20);(2,4)(13,14)(15,20)(16,18)(17,19);(5,7,8,12)(6,9,10,11)(14,18)(15,16)(19,20);(14,18)(15,16)(19,20);(14,19)(18,20)Polycyclic presentation:
2^9|a^2=g,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257750, #72257831, #72305401, … |
| maximal normal subgroup of | #72257750, #72257831, #72305401, … |
| maximal quotient of | #72264171, #72264290, C2×#6445476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352672);