Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1566 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)(9,12)(10,11)(13,14,16,18,17,20,15,19);(5,7,8,6,10,12,9,11)(14,18)(15,16)(19,20);(2,4)(14,18)(15,16)(19,20);(5,8,10,9)(6,7,11,12)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72310286, #72310287, #72310293, … |
| maximal normal subgroup of | #72310286, #72310287, #72310293, … |
| maximal quotient of | C2×#197726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104922);