Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213788 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13)(8,17)(9,12)(10,16)(11,18)(14,20)(15,19);(1,2)(3,4)(8,18)(11,17)(13,19)(16,20);(2,4)(5,7)(6,10,12,14)(8,16)(9,15)(11,13,17,19)(18,20);(5,6)(7,14)(8,11)(9,12)(10,15)(13,20)(16,19)(17,18)Polycyclic presentation:
2^9|c^2=h,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260388, #72260535, #72261611, … |
| maximal normal subgroup of | #72260388, #72260535, #72261611, … |
| maximal quotient of | #72238513 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289639);