Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20);(5,6)(7,13,15,10)(8,11)(9,12)(14,19,16,18)(17,20);(2,4)(5,7)(6,10)(8,16,17,14)(9,15)(11,19,20,18)(12,13);(5,8)(6,11)(7,14)(9,17)(10,18)(12,20)(13,19)(15,16)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=h,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289797);