Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5026 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)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(10,16)(17,20);(1,2,3,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,17,16,20)Polycyclic presentation:
2^9|c^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,f^c=fi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216108, #72227268, #72227280, … |
| maximal normal subgroup of | #72216108, #72227268, #72227280, … |
| maximal quotient of | #72219648, C2×#6442721, C4×#24260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349917);