Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444408 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3248 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)(5,6,8,13,10,14,9,11)(7,12,17,15,16,20,18,19);(2,4)(5,6)(7,15)(8,11)(9,13)(10,14)(12,17)(16,19)(18,20);(6,11,14,13)(12,19,20,15);(5,7,8,17,10,16,9,18)(6,12,13,15,14,20,11,19);(7,16)(12,20)(15,19)(17,18)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213250, #72213663, #72213751, … |
| maximal normal subgroup of | #72213250, #72213663, #72213751, … |
| maximal quotient of | #72216137, #72216140, #72216148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372878);