Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437445 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4897 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)(7,13)(8,16)(9,12)(10,15)(11,18)(14,20)(17,19);(5,7)(6,10)(8,17)(9,14)(11,20)(12,19)(13,16)(15,18);(1,2,3,4)(8,18)(10,19)(11,16)(15,17);(8,18)(10,19)(11,16)(15,17);(7,14)(10,19)(13,20)(15,17)Polycyclic presentation:
2^9|c^2=g,f^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,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239252, #72257360, #72257375, … |
| maximal normal subgroup of | #72239252, #72257360, #72257375, … |
| maximal quotient of | #72239276, #72257937, C2×#6442561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349757);