Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,15,11,7,16,14)(8,12,18,17,19,13,9,20);(5,7)(6,11)(8,17)(9,12)(10,14)(13,18)(15,16)(19,20);(1,2,3,4)(5,8,16,9,11,19,10,18)(6,12,15,17,7,13,14,20);(5,9,10,8,11,18,16,19)(6,13,15,20,7,12,14,17)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284132);