Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,9,13,11,16,10,15)(7,17,18,28,21,27,20,12)(8,14,23,29,25,32,24,31)(19,33,34,36,35,30,22,26);(2,4)(5,7)(6,12)(8,22)(9,20)(10,18)(11,21)(13,27)(14,30)(15,17)(16,28)(19,24)(23,35)(25,34)(26,31)(29,36)(32,33);(6,13,16,15)(7,18,21,20)(14,29,32,31)(19,34,35,22);(5,8,10,24,11,25,9,23)(6,14,15,31,16,32,13,29)(7,19,20,22,21,35,18,34)(12,26,27,30,28,36,17,33);(5,9,11,10)(6,13,16,15)(7,20,21,18)(8,23,25,24)(12,27,28,17)(14,29,32,31)(19,22,35,34)(26,30,36,33)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=h,e^2=i,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=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372871);