Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| 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 | |
3328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,12,10,8,19,13)(7,14,18,15,17,20,16,11);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,20)(13,14)(18,19);(2,4)(5,8)(6,10)(7,15)(9,13)(11,17)(12,19)(14,16)(18,20);(7,16)(9,19)(11,20)(12,13)(14,15)(17,18);(7,17)(11,15)(14,20)(16,18)Polycyclic presentation:
2^9|a^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213527, #72213751, #72213818, … |
| maximal normal subgroup of | #72213527, #72213751, #72213818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408078);