Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3537 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8,11,12)(9,10)(13,14,17,20,18,15,16,19);(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,20)(17,19);(5,8)(6,7)(9,12)(10,11)(16,17)(19,20);(13,16,18,17)(14,19,15,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309172, #72309174, #72309223, … |
| maximal normal subgroup of | #72309172, #72309174, #72309223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93225);