Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,10,11)(7,16,13,19)(8,17,18,12)(9,15,20,14);(1,2)(3,4)(5,7,10,13)(6,8,11,18)(9,17,20,12)(14,19,15,16);(2,4)(6,11)(7,13)(12,17)(14,15);(2,4)(5,8)(6,12)(7,15)(9,16)(10,18)(11,17)(13,14)(19,20);(5,8)(6,13)(7,11)(9,19)(10,18)(12,14)(15,17)(16,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72219568, #72219569, … |
| maximal normal subgroup of | #72213527, #72219568, #72219569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421346);