Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C22×D42 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(14,18)(15,16)(19,20);(1,2,3,4)(6,9)(10,11)(13,14,15,18,17,19,16,20);(5,7,8,12)(6,10,9,11)(14,18)(15,16)(19,20);(14,18)(15,16)(19,20);(14,19)(18,20)Polycyclic presentation:
2^9|b^2=fg,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^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72237567, #72237573, #72252514, … |
| maximal normal subgroup of | #72237567, #72237573, #72252514, … |
| maximal quotient of | #72257945, #72315917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353878);