Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439029 |
| 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 | |
3298 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)(7,14)(8,18)(9,12)(10,13)(11,17)(15,16)(19,20);(5,6,9,18,10,13,8,12)(7,15,19,20,16,14,17,11);(2,4)(5,7,8,17,10,16,9,19)(6,11,12,14,13,20,18,15);(5,6)(7,11)(8,12)(9,18)(10,13)(14,17)(15,19)(16,20);(7,16)(11,20)(14,15)(17,19)Polycyclic presentation:
2^9|b^2=fi,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72213590, #72213751, … |
| maximal normal subgroup of | #72213527, #72213590, #72213751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346227);